We study the asymptotic behaviour of the mean square of long truncations of the Dirichlet series for ( ζ ′ / ζ ) k (\zeta'/\zeta)^k ( ζ ′ / ζ ) k Λ ∗ k \Lambda^{*k} Λ ∗ k Λ ∗ k \Lambda^{*k} Λ ∗ k F q [ T ] \mathbb{F}_q[T] F q [ T ] δ k ( c ) \delta_k(c) δ k ( c ) c c c δ k ( c ) \delta_k(c) δ k ( c ) Λ ∗ k \Lambda^{*k} Λ ∗ k γ k ( c ) \gamma_k(c) γ k ( c ) ζ ( s ) k \zeta(s)^k ζ ( s ) k ( − ζ ′ / ζ ) k (-\zeta'/\zeta)^k ( − ζ ′ / ζ ) k
Introduction
This paper belongs to a series of works relating almost-prime statistics to moments of logarithmic derivatives of ζ ( s ) \zeta(s) ζ ( s ) Λ ∗ k \Lambda^{*k} Λ ∗ k [24 ] were studied in [20 ] , while here we recover the predicted leading coefficients for long Dirichlet polynomial mean squares and almost-prime variances over the integers, under shifted moment conjectures and RH. Forthcoming work [2 ] examines the exact finite-N N N τ \tau τ ζ ′ / ζ \zeta'/\zeta ζ ′ / ζ
Averages of long polynomials
The moments of the Riemann zeta function ζ ( s ) \zeta(s) ζ ( s ) ζ ( s ) \zeta(s) ζ ( s )
ζ ′ ζ ( s ) = − ∑ n = 1 ∞ Λ ( n ) n s , ℜ s > 1. (1) \frac{\zeta'}{\zeta}(s)=-\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n^s},\qquad\Re s>1. \tag*{(1)} ζ ζ ′ ( s ) = − n = 1 ∑ ∞ n s Λ ( n ) , ℜ s > 1. (1) Here, Λ ( n ) \Lambda(n) Λ ( n )
Λ ( n ) : = { log p ∃ m ∈ N , p ∈ P : n = p m 0 otherwise. (2) \Lambda(n):=
\begin{cases}
\log p & \exists m\in\mathbb{N},\ p\in\mathbb{P}: n=p^m\\
0 & \text{otherwise.}
\end{cases} \tag*{(2)} Λ ( n ) := { log p 0 ∃ m ∈ N , p ∈ P : n = p m otherwise. (2) This may be interpreted as a weighted counting function for prime powers. As the meromorphic continuation of ζ ′ / ζ ( s ) \zeta'/\zeta(s) ζ ′ / ζ ( s ) ζ ( s ) \zeta(s) ζ ( s ) a > 0 a>0 a > 0 [1 ] implies there exists a function c k : R + → R c_k:\mathbb{R}_{+}\to\mathbb{R} c k : R + → R
( 2 a ) 2 k − 1 T ∫ 0 T ∣ ζ ′ ζ ( 1 2 + a log T + i t ) ∣ 2 k d t ∼ c k ( a ) ( log T ) 2 k , k ∈ N , a > 0 , (3) \frac{(2a)^{2k-1}}{T}\int_{0}^{T}\left|\frac{\zeta'}{\zeta}\left(\frac{1}{2}+\frac{a}{\log T}+it\right)\right|^{2k}\,dt\sim c_k(a)(\log T)^{2k},\qquad k\in\mathbb{N},\quad a>0, \tag*{(3)} T ( 2 a ) 2 k − 1 ∫ 0 T ζ ζ ′ ( 2 1 + log T a + i t ) 2 k d t ∼ c k ( a ) ( log T ) 2 k , k ∈ N , a > 0 , (3) as T → ∞ T\to\infty T → ∞ k ∈ N k\in\mathbb{N} k ∈ N [8 ] form of the ratios conjecture [7 ] under RH. Furthermore, the same conjecture in [1 ] also predicts
c k ( a ) ∼ ( 2 k − 2 k − 1 ) , (4) c_k(a)\sim\binom{2k-2}{k-1}, \tag*{(4)} c k ( a ) ∼ ( k − 1 2 k − 2 ) , (4) as a → 0 + a\to0^{+} a → 0 + a a a (3) averaged over [ T , 2 T ] [T,2T] [ T , 2 T ] [14 ] under both RH and the 2 k 2k 2 k a a a [30 ] obtained an upper bound for the corresponding second moment in the range a ≥ 10 a\geq10 a ≥ 10 [11 ] investigated mixed shifted generalisations of the dyadic form of (3) , relating them to almost primes and higher correlations of the zeros of ζ ( s ) \zeta(s) ζ ( s ) [28 ] proved
1 T ∫ T 2 T ∣ ζ ′ ζ ( 1 2 + a log T + i t ) ∣ 2 k d t ≪ a , k ( log T ) 2 k , (5) \frac{1}{T}\int_{T}^{2T}\left|\frac{\zeta'}{\zeta}\left(\frac{1}{2}+\frac{a}{\log T}+it\right)\right|^{2k}\,dt\ll_{a,k}(\log T)^{2k}, \tag*{(5)} T 1 ∫ T 2 T ζ ζ ′ ( 2 1 + log T a + i t ) 2 k d t ≪ a , k ( log T ) 2 k , (5) for every fixed k ∈ N k\in\mathbb{N} k ∈ N a > 0 a>0 a > 0 (3) . A standard method for studying such moments is to replace the relevant Dirichlet series by a long Dirichlet polynomial and to study its mean values in t t t [3 ] who obtained precise asymptotics for mean squares of long Dirichlet polynomials, in their case for the mean square for ζ ( s ) k \zeta(s)^k ζ ( s ) k d k ( n ) d_k(n) d k ( n ) d k ( n ) d_k(n) d k ( n ) k k k d k ( n ) d_k(n) d k ( n ) n n n k k k
Explicitly, one has d k ( n ) = ( 1 ∗ ⋯ ∗ 1 ) ( n ) d_k(n)=(1*\cdots*1)(n) d k ( n ) = ( 1 ∗ ⋯ ∗ 1 ) ( n ) k k k ζ ( s ) \zeta(s) ζ ( s ) f f f g g g f ∗ g f*g f ∗ g
( f ∗ g ) ( n ) = ∑ d ∣ n f ( d ) g ( n d ) , (f*g)(n)=\sum_{d\mid n}f(d)g\left(\frac{n}{d}\right), ( f ∗ g ) ( n ) = d ∣ n ∑ f ( d ) g ( d n ) , where the sum extends over all positive divisors d d d n n n ∗ * ∗
In studying the moments
1 T ∫ 0 T ∣ ζ ′ ζ ( 1 2 + κ + i t ) ∣ 2 k d t , (6) \frac{1}{T}\int_{0}^{T}\left|\frac{\zeta'}{\zeta}\left(\frac{1}{2}+\kappa+it\right)\right|^{2k}\,dt, \tag*{(6)} T 1 ∫ 0 T ζ ζ ′ ( 2 1 + κ + i t ) 2 k d t , (6) for κ = a log T \kappa=\frac{a}{\log T} κ = l o g T a a > 0 a>0 a > 0 ( − ζ ′ / ζ ( s ) ) k \left(-\zeta'/\zeta(s)\right)^k ( − ζ ′ / ζ ( s ) ) k k k k Λ ( n ) \Lambda(n) Λ ( n ) k ∈ N k\in\mathbb{N} k ∈ N
Λ ∗ k : = Λ ∗ ⋯ ∗ Λ ⏟ k times , (7) \Lambda^{*k}:=\underbrace{\Lambda*\cdots*\Lambda}_{k\ \text{times}}, \tag*{(7)} Λ ∗ k := k times Λ ∗ ⋯ ∗ Λ , (7) and the Dirichlet series
( − ζ ′ ζ ( s ) ) k = ∑ n = 1 ∞ Λ ∗ k ( n ) n s , ( ℜ s > 1 ) . (8) \left(-\frac{\zeta'}{\zeta}(s)\right)^k=\sum_{n=1}^{\infty}\frac{\Lambda^{*k}(n)}{n^s},\qquad(\Re s>1). \tag*{(8)} ( − ζ ζ ′ ( s ) ) k = n = 1 ∑ ∞ n s Λ ∗ k ( n ) , ( ℜ s > 1 ) . (8) Let N ≥ 1 N \ge1 N ≥ 1 s ∈ C s \in\mathbb{C} s ∈ C σ > max ( 1 , ℜ s ) \sigma> \max(1,\Re s) σ > max ( 1 , ℜ s )
∑ n ≤ N ′ Λ ∗ k ( n ) n s = 1 2 π i ∫ ( σ ) ( − ζ ′ ζ ( w ) ) k N w − s w − s d w . (9) \sum_{n \le N}^{\prime} \frac{\Lambda^{*k}(n)}{n^s} = \frac{1}{2\pi i}\int_{(\sigma)}\left(-\frac{\zeta'}{\zeta}(w)\right)^k\frac{N^{w-s}}{w-s}\,dw. \tag*{(9)} n ≤ N ∑ ′ n s Λ ∗ k ( n ) = 2 π i 1 ∫ ( σ ) ( − ζ ζ ′ ( w ) ) k w − s N w − s d w . (9) The prime on the sum denotes the endpoint term n = N n=N n = N 1 / 2 1/2 1/2 N N N ( σ ) (\sigma) ( σ ) ℜ w = σ \Re w=\sigma ℜ w = σ w w w w = 1 w=1 w = 1
Res w = 1 ( ( − ζ ′ ζ ( w ) ) k N w − s w − s ) . (10) \operatorname{Res}_{w=1}\left(\left(-\frac{\zeta'}{\zeta}(w)\right)^k\frac{N^{w-s}}{w-s}\right). \tag*{(10)} Res w = 1 ( ( − ζ ζ ′ ( w ) ) k w − s N w − s ) . (10) Motivated by (3) , we set s = 1 2 + i t s=\frac{1}{2}+it s = 2 1 + i t
I k ( T , N ) : = 1 T ∫ 0 T ∣ ∑ 1 ≤ n ≤ N Λ ∗ k ( n ) n 1 / 2 + i t − Res w = 1 ( ( − ζ ′ ζ ( w ) ) k N w − 1 2 − i t w − 1 2 − i t ) ∣ 2 d t , (11) I_k(T,N):=\frac{1}{T}\int_0^T\left|\sum_{1\le n\le N}\frac{\Lambda^{*k}(n)}{n^{1/2+it}}-\operatorname{Res}_{w=1}\left(\left(-\frac{\zeta'}{\zeta}(w)\right)^k\frac{N^{w-\frac{1}{2}-it}}{w-\frac{1}{2}-it}\right)\right|^2dt, \tag*{(11)} I k ( T , N ) := T 1 ∫ 0 T 1 ≤ n ≤ N ∑ n 1/2 + i t Λ ∗ k ( n ) − Res w = 1 ( ( − ζ ζ ′ ( w ) ) k w − 2 1 − i t N w − 2 1 − i t ) 2 d t , (11) where N = T α N=T^\alpha N = T α α ∈ R + \alpha\in\mathbb{R}_+ α ∈ R + (6) , I k ( T , T α ) I_k(T,T^\alpha) I k ( T , T α ) (3) , for any fixed α > 0 \alpha>0 α > 0 I k ( T , T α ) I_k(T,T^\alpha) I k ( T , T α ) ( log T ) 2 k (\log T)^{2k} ( log T ) 2 k T → ∞ T\to\infty T → ∞ [1 ] . This leads us to introduce
M k ( α ) : = lim T → ∞ I k ( T , T α ) ( log T ) 2 k , k ∈ N , α > 0 , (12) M_k(\alpha):=\lim_{T\to\infty}\frac{I_k(T,T^\alpha)}{(\log T)^{2k}},\qquad k\in\mathbb{N},\ \alpha>0, \tag*{(12)} M k ( α ) := T → ∞ lim ( log T ) 2 k I k ( T , T α ) , k ∈ N , α > 0 , (12) whenever the limit exists. Under Conjecture 1, we will prove that M k ( α ) M_k(\alpha) M k ( α ) ζ ( s ) k \zeta(s)^k ζ ( s ) k [3 ] .
As an aside for readers familiar with the associated literature, the mean-square problem for the case k = 1 k=1 k = 1 F ( u , T ) F(u,T) F ( u , T ) [27 ] represents these differences by a normalised, weighted Fourier sum. Under RH, Goldston, Gonek and Montgomery [16 ] related the shifted second moment of ζ ′ / ζ ( s ) \zeta'/\zeta(s) ζ ′ / ζ ( s ) F ( u , T ) F(u,T) F ( u , T ) [15 ] proved, also under RH, that the strong pair-correlation conjecture is equivalent to suitably uniform asymptotics for the variance of primes in short intervals. Farmer [12 ] extended this perspective beyond pair correlation, proving an asymptotic for a third-order mixed moment of ζ ′ / ζ ( s ) \zeta'/\zeta(s) ζ ′ / ζ ( s ) [4 ] later obtained improved RH-conditional bounds for the second moment of ζ ′ / ζ ( s ) \zeta'/\zeta(s) ζ ′ / ζ ( s ) F ( u , T ) F(u,T) F ( u , T ) [5 ] also extended the Goldston–Gonek–Montgomery equivalence to second moments of the derivatives ( ζ ′ / ζ ) ( j ) (\zeta'/\zeta)^{(j)} ( ζ ′ / ζ ) ( j ) j ≥ 1 j\ge1 j ≥ 1
Due to the cumulative dependence on α \alpha α T α z / z T^{\alpha z}/z T α z / z
T α z − 1 z = ( log T ) ∫ 0 α T u z d u , (13) \frac{T^{\alpha z}-1}{z}=(\log T)\int_0^\alpha T^{uz}\,du, \tag*{(13)} z T α z − 1 = ( log T ) ∫ 0 α T u z d u , (13) we may take at a heuristic level that
M 1 ( α ) = ∫ 0 α f ( u ) d u , (14) M_1(\alpha)=\int_0^\alpha f(u)\,du, \tag*{(14)} M 1 ( α ) = ∫ 0 α f ( u ) d u , (14) where f ( u ) f(u) f ( u ) F ( u , T ) F(u,T) F ( u , T ) T → ∞ T\to\infty T → ∞ u = 0 u=0 u = 0 f ( u ) = u f(u)=u f ( u ) = u 0 < u < 1 0<u<1 0 < u < 1 f ( u ) = 1 f(u)=1 f ( u ) = 1 u ≥ 1 u\ge1 u ≥ 1 [27 ] ; hence
M 1 ( α ) = { α 2 2 , 0 < α < 1 , α − 1 2 , α ≥ 1 , (15) M_1(\alpha)=\begin{cases}\frac{\alpha^2}{2}, & 0<\alpha<1,\\ \alpha-\frac{1}{2}, & \alpha\ge1,\end{cases} \tag*{(15)} M 1 ( α ) = { 2 α 2 , α − 2 1 , 0 < α < 1 , α ≥ 1 , (15) under both assumptions. No explicit description appears to be available in the literature when k ≥ 2 k\ge2 k ≥ 2 M k ( α ) M_k(\alpha) M k ( α ) [1 ] , equation 43) and we also derive an unsymmetrised version of this expression, given in equation (104) in § 3.4 . Based on the observed structure in these results, we conjecture the following form for shifted moments of the logarithmic derivative of ζ ( s ) \zeta(s) ζ ( s )
Conjecture 1. Let k ∈ N k\in\mathbb{N} k ∈ N Y = Y ( T ) ≥ 1 Y=Y(T)\ge1 Y = Y ( T ) ≥ 1 T > 1 T>1 T > 1 ℜ z 1 > 0 \Re z_1>0 ℜ z 1 > 0 0 < ξ < 1 0<\xi<1 0 < ξ < 1
1 T ∫ 0 T ( ζ ′ ζ ( 1 2 + i t + z 1 ) ) k ( ζ ′ ζ ( 1 2 − i t − z 2 ) ) k d t (16) \frac{1}{T}\int_0^T\left(\frac{\zeta'}{\zeta}\left(\frac{1}{2}+it+z_1\right)\right)^k\left(\frac{\zeta'}{\zeta}\left(\frac{1}{2}-it-z_2\right)\right)^k\,dt \tag*{(16)} T 1 ∫ 0 T ( ζ ζ ′ ( 2 1 + i t + z 1 ) ) k ( ζ ζ ′ ( 2 1 − i t − z 2 ) ) k d t (16) = 1 k ! 2 ( 2 π i ) 2 k ∫ C u k ∫ C v k T 1 2 ( ∑ j = 1 k u j − ∑ ℓ = 1 k v ℓ ) ∏ j , ℓ = 1 k ( v ℓ − u j − z ) ∏ 1 ≤ i < j ≤ k ( u i − u j ) 2 ∏ 1 ≤ i < j ≤ k ( v i − v j ) 2 =\frac{1}{k!^2(2\pi i)^{2k}}\int_{C_u^k}\int_{C_v^k}T^{\frac{1}{2}(\sum_{j=1}^k u_j-\sum_{\ell=1}^k v_\ell)}
\prod_{j,\ell=1}^k(v_\ell-u_j-z)
\prod_{1\le i<j\le k}(u_i-u_j)^2
\prod_{1\le i<j\le k}(v_i-v_j)^2 = k ! 2 ( 2 π i ) 2 k 1 ∫ C u k ∫ C v k T 2 1 ( ∑ j = 1 k u j − ∑ ℓ = 1 k v ℓ ) j , ℓ = 1 ∏ k ( v ℓ − u j − z ) 1 ≤ i < j ≤ k ∏ ( u i − u j ) 2 1 ≤ i < j ≤ k ∏ ( v i − v j ) 2 × ( ∑ j = 1 k 1 u j + ∑ ℓ = 1 k 1 v ℓ − z − log T 2 ) k ( ∑ j = 1 k 1 u j + z + ∑ ℓ = 1 k 1 v ℓ + log T 2 ) k z − k 2 \qquad{}\times\left(\sum_{j=1}^k\frac{1}{u_j}+\sum_{\ell=1}^k\frac{1}{v_\ell-z}-\frac{\log T}{2}\right)^k
\left(\sum_{j=1}^k\frac{1}{u_j+z}+\sum_{\ell=1}^k\frac{1}{v_\ell}+\frac{\log T}{2}\right)^k z^{-k^2} × ( j = 1 ∑ k u j 1 + ℓ = 1 ∑ k v ℓ − z 1 − 2 log T ) k ( j = 1 ∑ k u j + z 1 + ℓ = 1 ∑ k v ℓ 1 + 2 log T ) k z − k 2 × ∏ j = 1 k u j − k d u j ∏ ℓ = 1 k v ℓ − k d v ℓ + O k , c , ξ ( ( log T ) 2 k − 1 + ξ ( 1 + min ( ∣ z ∣ , 1 ) log T ) − θ k , c ) , \qquad{}\times\prod_{j=1}^k u_j^{-k}\,du_j\prod_{\ell=1}^k v_\ell^{-k}\,dv_\ell
+O_{k,c,\xi}\left((\log T)^{2k-1+\xi}(1+\min(|z|,1)\log T)^{-\theta_{k,c}}\right), × j = 1 ∏ k u j − k d u j ℓ = 1 ∏ k v ℓ − k d v ℓ + O k , c , ξ ( ( log T ) 2 k − 1 + ξ ( 1 + min ( ∣ z ∣ , 1 ) log T ) − θ k , c ) , where z = z 1 − z 2 z=z_1-z_2 z = z 1 − z 2 ∣ ℑ z ∣ , ∣ ℑ z 2 ∣ ≤ Y |\Im z|,|\Im z_2|\le Y ∣ℑ z ∣ , ∣ℑ z 2 ∣ ≤ Y 0 < − ℜ z 2 , ℜ z ≤ 3 log T 0<-\Re z_2,\Re z\le\frac{3}{\log T} 0 < − ℜ z 2 , ℜ z ≤ l o g T 3 C u = C u ( z ) C_u=C_u(z) C u = C u ( z ) C v = C v ( z ) C_v=C_v(z) C v = C v ( z ) C u C_u C u − z -z − z C v C_v C v z z z θ k , c : = k − 1 \theta_{k,c}:=k-1 θ k , c := k − 1 k ≥ 3 k\ge3 k ≥ 3 θ k , c : = 1 + c \theta_{k,c}:=1+c θ k , c := 1 + c k = 1 , 2 k=1,2 k = 1 , 2 c > 0 c>0 c > 0 k = 1 , 2 k=1,2 k = 1 , 2 0 < ξ < c ≤ 1 0<\xi<c\le1 0 < ξ < c ≤ 1
As detailed in the proof of Theorem 2.5 , one can also differentiate the contour integral form of the shifted ratios conjecture (stated in [7 ] , equation 6.22) to obtain the above form. In fact, this approach suggests a stronger error term of O k , ε ( T − 1 2 + ε ( log T ) 2 k ) O_{k,\varepsilon}(T^{-\frac{1}{2}+\varepsilon}(\log T)^{2k}) O k , ε ( T − 2 1 + ε ( log T ) 2 k )
Averages of von Mangoldt convolutions
The asymptotic behaviour of the variance of sums of Dirichlet coefficients is also the subject of extensive study [18 , 25 ] , in part due to its connection to moment problems for ζ ( s ) \zeta(s) ζ ( s ) L L L [3 ] , Bettin and Conrey analyse the variance of the divisor function in short intervals through its relationship with moments of ζ ( s ) k \zeta(s)^k ζ ( s ) k k k k Λ ∗ k ( n ) \Lambda^{*k}(n) Λ ∗ k ( n ) ( − ζ ′ / ζ ( s ) ) k (-\zeta'/\zeta(s))^k ( − ζ ′ / ζ ( s ) ) k x , H > 0 x,H>0 x , H > 0 x ∈ [ X , 2 X ] x\in[X,2X] x ∈ [ X , 2 X ]
Ψ k ( x ; H ) : = ∑ x < n ≤ x + H Λ ∗ k ( n ) , (17) \Psi_k(x;H):=\sum_{x<n\le x+H}\Lambda^{*k}(n), \tag*{(17)} Ψ k ( x ; H ) := x < n ≤ x + H ∑ Λ ∗ k ( n ) , (17) for which we are interested in the short-interval regime H = o ( X ) H=o(X) H = o ( X ) ( − ζ ′ / ζ ( s ) ) k \left(-\zeta'/\zeta(s)\right)^k ( − ζ ′ / ζ ( s ) ) k s = 1 s=1 s = 1
Δ k ( x ; H ) : = Ψ k ( x ; H ) − Res s = 1 [ ( − ζ ′ ζ ( s ) ) k ( x + H ) s − x s s ] . (18) \Delta_k(x;H):=\Psi_k(x;H)-\operatorname*{Res}_{s=1}\left[\left(-\frac{\zeta'}{\zeta}(s)\right)^k\frac{(x+H)^s-x^s}{s}\right]. \tag*{(18)} Δ k ( x ; H ) := Ψ k ( x ; H ) − s = 1 Res [ ( − ζ ζ ′ ( s ) ) k s ( x + H ) s − x s ] . (18) We consider
V k ( X , H ) = 1 X H ∫ X 2 X ∣ Δ k ( x ; H ) ∣ 2 d x (19) \mathcal{V}_k(X,H)=\frac{1}{XH}\int_X^{2X}\left|\Delta_k(x;H)\right|^2\,dx \tag*{(19)} V k ( X , H ) = X H 1 ∫ X 2 X ∣ Δ k ( x ; H ) ∣ 2 d x (19) for H = X 1 − 1 / α H=X^{1-1/\alpha} H = X 1 − 1/ α α > 0 \alpha>0 α > 0 k = 1 k=1 k = 1 [15 ] . For general k ∈ N k\in\mathbb{N} k ∈ N ( − ζ ′ / ζ ( s ) ) k \left(-\zeta'/\zeta(s)\right)^k ( − ζ ′ / ζ ( s ) ) k [11 ] , as an integer n n n k k k α > 0 \alpha>0 α > 0 k ∈ N k\in\mathbb{N} k ∈ N
D k ( α ) : = lim X → ∞ V k ( X , X 1 − 1 / α ) ( log X 1 / α ) 2 k − 1 . (20) \mathcal{D}_k(\alpha):=\lim_{X\to\infty}\frac{\mathcal{V}_k\left(X,X^{1-1/\alpha}\right)}{\left(\log X^{1/\alpha}\right)^{2k-1}}. \tag*{(20)} D k ( α ) := X → ∞ lim ( log X 1/ α ) 2 k − 1 V k ( X , X 1 − 1/ α ) . (20) Bettin and Conrey [3 ] showed that, in the case of d k ( n ) d_k(n) d k ( n ) M k ( α ) M_k(\alpha) M k ( α ) ( − ζ ′ / ζ ( s ) ) k \left(-\zeta'/\zeta(s)\right)^k ( − ζ ′ / ζ ( s ) ) k Λ ∗ k ( n ) \Lambda^{*k}(n) Λ ∗ k ( n ) Theorem 2.3 .
Averages of von Mangoldt convolutions in arithmetic progressions
We now examine this problem in arithmetic progressions. For q ∈ N q\in\mathbb{N} q ∈ N X > 0 X>0 X > 0
W k ( X ; q ) : = 1 X ∑ 1 ≤ a ≤ q ( a , q ) = 1 ∣ ∑ n ≤ X n ≡ a ( m o d q ) Λ ∗ k ( n ) − 1 φ ( q ) Res s = 1 ( ( − ζ q ′ ζ q ( s ) ) k X s s ) ∣ 2 , (21) \mathcal{W}_k(X;q):=\frac{1}{X}\sum_{\substack{1\leq a\leq q\\(a,q)=1}}\left|\sum_{\substack{n\leq X\\n\equiv a\pmod q}}\Lambda^{*k}(n)-\frac{1}{\varphi(q)}\operatorname*{Res}_{s=1}\left(\left(-\frac{\zeta_q'}{\zeta_q}(s)\right)^k\frac{X^s}{s}\right)\right|^2, \tag*{(21)} W k ( X ; q ) := X 1 1 ≤ a ≤ q ( a , q ) = 1 ∑ n ≤ X n ≡ a ( mod q ) ∑ Λ ∗ k ( n ) − φ ( q ) 1 s = 1 Res ( ( − ζ q ζ q ′ ( s ) ) k s X s ) 2 , (21) where ζ q ( s ) : = ∑ n ≥ 1 ( n , q ) = 1 n − s \zeta_q(s):=\sum_{\substack{n\geq1\\(n,q)=1}}n^{-s} ζ q ( s ) := ∑ n ≥ 1 ( n , q ) = 1 n − s φ \varphi φ φ \varphi φ W k ( X ; q ) \mathcal{W}_k(X;q) W k ( X ; q ) W k ( X , q ) \mathcal{W}_k(X,q) W k ( X , q )
Q k ( α ) : = lim q → ∞ W k ( X ; q ) ( log q ) 2 k − 1 , α > 0 , (22) \mathcal{Q}_k(\alpha):=\lim_{q\to\infty}\frac{\mathcal{W}_k(X;q)}{(\log q)^{2k-1}},\qquad\alpha>0, \tag*{(22)} Q k ( α ) := q → ∞ lim ( log q ) 2 k − 1 W k ( X ; q ) , α > 0 , (22) whenever the limit exists, where q q q X = X ( q ) X=X(q) X = X ( q ) log X / log q → α \log X/\log q\to\alpha log X / log q → α q = p q=p q = p k = 1 k=1 k = 1 ψ ( X ; q , a ) \psi(X;q,a) ψ ( X ; q , a ) [13 ] . They obtained a lower bound for k = 1 k=1 k = 1 k k k k > 1 k>1 k > 1 [23 ] proved lower bounds for the variance associated with the generalised von Mangoldt function Λ k = μ ∗ ( log ) k \Lambda_k=\mu*(\log)^k Λ k = μ ∗ ( log ) k μ \mu μ k k k Λ ∗ k ( n ) \Lambda^{*k}(n) Λ ∗ k ( n )
For a Dirichlet character χ \chi χ q q q L ( s , χ ) = ∑ n = 1 ∞ χ ( n ) n − s L(s,\chi)=\sum_{n=1}^{\infty}\chi(n)n^{-s} L ( s , χ ) = ∑ n = 1 ∞ χ ( n ) n − s ℜ s > 1 \Re s>1 ℜ s > 1 L L L Q k ( α ) \mathcal{Q}_k(\alpha) Q k ( α ) L L L t t t
Conjecture 2. Let q q q k ∈ N k \in\mathbb{N} k ∈ N Y = Y ( q ) ≥ 1 Y=Y(q) \ge1 Y = Y ( q ) ≥ 1 ℜ z 1 > 0 \Re z_1 > 0 ℜ z 1 > 0 0 < ξ < 1 0 < \xi< 1 0 < ξ < 1 q → ∞ q \to\infty q → ∞
1 φ ( q ) ∑ χ m o d q ( L ′ L ( 1 2 + z 1 , χ ) ) k ( L ′ L ( 1 2 − z 2 , χ ‾ ) ) k = 1 k ! 2 ( 2 π i ) 2 k ∫ C u k ∫ C v k q 1 2 ( ∑ j = 1 k u j − ∑ ℓ = 1 k v ℓ ) ∏ 1 ≤ i < j ≤ k ( u i − u j ) 2 ∏ 1 ≤ i < j ≤ k ( v i − v j ) 2 \frac{1}{\varphi(q)}\sum_{\chi\bmod q}\left(\frac{L'}{L}\left(\frac{1}{2}+z_1,\chi\right)\right)^k\left(\frac{L'}{L}\left(\frac{1}{2}-z_2,\overline{\chi}\right)\right)^k
= \frac{1}{k!^2(2\pi i)^{2k}}\int_{C_u^k}\int_{C_v^k}q^{\frac{1}{2}\left(\sum_{j=1}^k u_j-\sum_{\ell=1}^k v_\ell\right)}
\prod_{1\le i<j\le k}(u_i-u_j)^2\prod_{1\le i<j\le k}(v_i-v_j)^2 φ ( q ) 1 χ mod q ∑ ( L L ′ ( 2 1 + z 1 , χ ) ) k ( L L ′ ( 2 1 − z 2 , χ ) ) k = k ! 2 ( 2 π i ) 2 k 1 ∫ C u k ∫ C v k q 2 1 ( ∑ j = 1 k u j − ∑ ℓ = 1 k v ℓ ) 1 ≤ i < j ≤ k ∏ ( u i − u j ) 2 1 ≤ i < j ≤ k ∏ ( v i − v j ) 2 × ( ∑ j = 1 k 1 u j + ∑ ℓ = 1 k 1 v ℓ − z − log q 2 ) k ( ∑ j = 1 k 1 u j + z + ∑ ℓ = 1 k 1 v ℓ + log q 2 ) k ∏ j , ℓ = 1 k ( v ℓ − u j − z ) {}\times\left(\sum_{j=1}^k\frac{1}{u_j}+\sum_{\ell=1}^k\frac{1}{v_\ell-z}-\frac{\log q}{2}\right)^k
\left(\sum_{j=1}^k\frac{1}{u_j+z}+\sum_{\ell=1}^k\frac{1}{v_\ell}+\frac{\log q}{2}\right)^k
\prod_{j,\ell=1}^k(v_\ell-u_j-z) × ( j = 1 ∑ k u j 1 + ℓ = 1 ∑ k v ℓ − z 1 − 2 log q ) k ( j = 1 ∑ k u j + z 1 + ℓ = 1 ∑ k v ℓ 1 + 2 log q ) k j , ℓ = 1 ∏ k ( v ℓ − u j − z ) × z − k 2 ∏ j = 1 k u j − k d u j ∏ ℓ = 1 k v ℓ − k d v ℓ + O k , ξ ( ( log q ) 2 k − 1 + ξ ( 1 + min ( ∣ z ∣ , 1 ) log q ) − θ k , c ) , (23) {}\times z^{-k^2}\prod_{j=1}^k u_j^{-k}\,du_j\prod_{\ell=1}^k v_\ell^{-k}\,dv_\ell
+O_{k,\xi}\left((\log q)^{2k-1+\xi}(1+\min(|z|,1)\log q)^{-\theta_{k,c}}\right), \tag*{(23)} × z − k 2 j = 1 ∏ k u j − k d u j ℓ = 1 ∏ k v ℓ − k d v ℓ + O k , ξ ( ( log q ) 2 k − 1 + ξ ( 1 + min ( ∣ z ∣ , 1 ) log q ) − θ k , c ) , (23) where z = z 1 − z 2 z=z_1-z_2 z = z 1 − z 2 ∣ ℑ z ∣ , ∣ ℑ z 2 ∣ ≤ Y |\Im z|,|\Im z_2|\le Y ∣ℑ z ∣ , ∣ℑ z 2 ∣ ≤ Y 0 < − ℜ z 2 , ℜ z ≤ 3 log q 0<-\Re z_2,\Re z\le\frac{3}{\log q} 0 < − ℜ z 2 , ℜ z ≤ l o g q 3 C u = C u ( z ) C_u=C_u(z) C u = C u ( z ) C v = C v ( z ) C_v=C_v(z) C v = C v ( z ) C u C_u C u 0 0 0 − z -z − z C v C_v C v 0 0 0 z z z θ k , c \theta_{k,c} θ k , c Conjecture 1 .
Random Matrix Theory
A central bridge in the study of the Riemann zeta function is the analogy with characteristic polynomials of random unitary matrices, which provides a framework for predicting the moments of ζ ( s ) \zeta(s) ζ ( s )
Over the polynomial ring F q [ t ] \mathbb{F}_q[t] F q [ t ] [22 ] studied the variance of short-interval sums of d k d_k d k q q q [22 Theorem 1.2] . Using similar methods, Kuperberg and Lalín [24 ] prove that the matrix integral arising from the variance of short-interval sums of Λ ∗ k \Lambda^{*k} Λ ∗ k [24 Theorem 1.3] . Such matrix integrals are closely related to moments of characteristic polynomials and their derivatives, as we make explicit below for the Λ ∗ k \Lambda^{*k} Λ ∗ k
Let U ∈ U ( N ) U \in U(N) U ∈ U ( N ) U U U e i θ 1 , … , e i θ N e^{i\theta_1},\ldots,e^{i\theta_N} e i θ 1 , … , e i θ N θ 1 , … , θ N ∈ [ 0 , 2 π ) \theta_1,\ldots,\theta_N \in[0,2\pi) θ 1 , … , θ N ∈ [ 0 , 2 π ) U U U
Λ U ( z ) : = det ( I − z U ∗ ) = ∏ n = 1 N ( 1 − z e − i θ n ) , (24) \Lambda_U(z):=\det(I-zU^*)=\prod_{n=1}^{N}(1-ze^{-i\theta_n}), \tag*{(24)} Λ U ( z ) := det ( I − z U ∗ ) = n = 1 ∏ N ( 1 − z e − i θ n ) , (24) with U ∗ U^* U ∗ U U U ∣ z ∣ < 1 |z|<1 ∣ z ∣ < 1 Λ U \Lambda_U Λ U
Λ U ′ ( z ) Λ U ( z ) = − ∑ n = 1 N e − i θ n 1 − z e − i θ n = − 1 z ∑ j = 1 ∞ z j Tr ( U ∗ j ) , (25) \frac{\Lambda_U'(z)}{\Lambda_U(z)}=-\sum_{n=1}^{N}\frac{e^{-i\theta_n}}{1-ze^{-i\theta_n}}=-\frac{1}{z}\sum_{j=1}^{\infty}z^j\operatorname{Tr}(U^{*j}), \tag*{(25)} Λ U ( z ) Λ U ′ ( z ) = − n = 1 ∑ N 1 − z e − i θ n e − i θ n = − z 1 j = 1 ∑ ∞ z j Tr ( U ∗ j ) , (25) where Tr ( U ∗ j ) : = ∑ n = 1 N e − i j θ n \operatorname{Tr}(U^{*j}):=\sum_{n=1}^{N}e^{-ij\theta_n} Tr ( U ∗ j ) := ∑ n = 1 N e − ij θ n U ∗ j U^{*j} U ∗ j Λ U ( z ) \Lambda_U(z) Λ U ( z ) ∣ x ∣ , ∣ y ∣ < 1 |x|,|y|<1 ∣ x ∣ , ∣ y ∣ < 1
F k ( x y ; N ) : = ∫ U ( N ) ( − x Λ U ′ ( x ) Λ U ( x ) ) k ( − y Λ U ∗ ′ ( y ) Λ U ∗ ( y ) ) k d U (26) F_k(xy;N):=\int_{U(N)}\left(-x\frac{\Lambda_U'(x)}{\Lambda_U(x)}\right)^k\left(-y\frac{\Lambda_{U^*}'(y)}{\Lambda_{U^*}(y)}\right)^k\,dU \tag*{(26)} F k ( x y ; N ) := ∫ U ( N ) ( − x Λ U ( x ) Λ U ′ ( x ) ) k ( − y Λ U ∗ ( y ) Λ U ∗ ′ ( y ) ) k d U (26) = ∑ n = k ∞ ( x y ) n I k ( n ; N ) . (27) =\sum_{n=k}^{\infty}(xy)^n I_k(n;N). \tag*{(27)} = n = k ∑ ∞ ( x y ) n I k ( n ; N ) . (27) where
I k ( n ; N ) = ∫ U ( N ) ∣ ∑ i 1 + ⋯ + i k = n i 1 , … , i k ≥ 1 Tr ( U i 1 ) ⋯ Tr ( U i k ) ∣ 2 d U . (28) I_k(n;N)=\int_{U(N)}\left|\sum_{\substack{i_1+\cdots+i_k=n\\i_1,\ldots,i_k\geq1}}\operatorname{Tr}(U^{i_1})\cdots\operatorname{Tr}(U^{i_k})\right|^2dU. \tag*{(28)} I k ( n ; N ) = ∫ U ( N ) i 1 + ⋯ + i k = n i 1 , … , i k ≥ 1 ∑ Tr ( U i 1 ) ⋯ Tr ( U i k ) 2 d U . (28) Here, d U dU d U (28) with matrix size N = n − h − 2 N=n-h-2 N = n − h − 2 n n n h h h α > 0 \alpha>0 α > 0
n = ⌊ α N ⌋ . n=\lfloor\alpha N\rfloor. n = ⌊ α N ⌋ . The floor ⌊ ⋅ ⌋ \lfloor\cdot\rfloor ⌊ ⋅ ⌋ n n n
δ k ( α ) : = lim N → ∞ I k ( ⌊ α N ⌋ ; N ) N 2 k − 1 , α > 0 , (29) \delta_k(\alpha):=\lim_{N\to\infty}\frac{I_k(\lfloor\alpha N\rfloor;N)}{N^{2k-1}},\qquad\alpha>0, \tag*{(29)} δ k ( α ) := N → ∞ lim N 2 k − 1 I k (⌊ α N ⌋ ; N ) , α > 0 , (29) whenever the limit exists. Note that expanding the k k k (26) produces ordered k k k ( i 1 , … , i k ) (i_1,\ldots,i_k) ( i 1 , … , i k ) ( j 1 , … , j k ) (j_1,\ldots,j_k) ( j 1 , … , j k ) i a , j b ≥ 1 i_a,j_b\geq1 i a , j b ≥ 1 U → e i φ U U\to e^{i\varphi}U U → e i φ U φ ∈ [ 0 , 2 π ) \varphi\in[0,2\pi) φ ∈ [ 0 , 2 π ) ∑ a = 1 k i a = ∑ b = 1 k j b = : n \sum_{a=1}^k i_a=\sum_{b=1}^k j_b=:n ∑ a = 1 k i a = ∑ b = 1 k j b =: n (27) . We define the cumulative, or cut-off, trace average
I ~ k ( n ; N ) : = ∫ U ( N ) ∣ ∑ j 1 + ⋯ + j k ≤ n j 1 , … , j k ≥ 1 Tr ( U j 1 ) ⋯ Tr ( U j k ) ∣ 2 d U . (30) \widetilde{I}_k(n;N):=\int_{U(N)}\left|\sum_{\substack{j_1+\cdots+j_k\leq n\\j_1,\ldots,j_k\geq1}}\operatorname{Tr}(U^{j_1})\cdots\operatorname{Tr}(U^{j_k})\right|^2dU. \tag*{(30)} I k ( n ; N ) := ∫ U ( N ) j 1 + ⋯ + j k ≤ n j 1 , … , j k ≥ 1 ∑ Tr ( U j 1 ) ⋯ Tr ( U j k ) 2 d U . (30) We may relate these matrix integrals through
I ~ k ( n ; N ) = ∑ m = k n I k ( m , N ) . (31) \widetilde{I}_k(n;N)=\sum_{m=k}^{n}I_k(m,N). \tag*{(31)} I k ( n ; N ) = m = k ∑ n I k ( m , N ) . (31) Therefore
∑ n ≥ k I ~ k ( n ; N ) t n = 1 1 − t ∑ m ≥ k I k ( m , N ) t m = F k ( t ; N ) 1 − t , (32) \sum_{n\geq k}\widetilde{I}_k(n;N)t^n=\frac{1}{1-t}\sum_{m\geq k}I_k(m,N)t^m=\frac{F_k(t;N)}{1-t}, \tag*{(32)} n ≥ k ∑ I k ( n ; N ) t n = 1 − t 1 m ≥ k ∑ I k ( m , N ) t m = 1 − t F k ( t ; N ) , (32) and consequently,
I ~ k ( n ; N ) = 1 2 π i ∮ ∣ t ∣ = r F k ( t ; N ) ( 1 − t ) t n + 1 d t , (33) \widetilde{I}_k(n;N)=\frac{1}{2\pi i}\oint_{|t|=r}\frac{F_k(t;N)}{(1-t)t^{n+1}}\,dt, \tag*{(33)} I k ( n ; N ) = 2 π i 1 ∮ ∣ t ∣ = r ( 1 − t ) t n + 1 F k ( t ; N ) d t , (33) for any 0 < r < 1 0<r<1 0 < r < 1 t = e − s / N t=e^{-s/N} t = e − s / N s = σ + i θ s=\sigma+i\theta s = σ + i θ r = e − σ / N r=e^{-\sigma/N} r = e − σ / N ∣ t ∣ = r |t|=r ∣ t ∣ = r
− π N ≤ θ ≤ π N . -\pi N\leq\theta\leq\pi N. − π N ≤ θ ≤ π N . In this case, the Cauchy coefficient extraction becomes a discrete Laplace inversion
I ~ k ( n ; N ) = 1 2 π i N ∫ σ − i π N σ + i π N e n s / N F k ( e − s / N ; N ) 1 − e − s / N d s . (34) \widetilde{I}_k(n;N)=\frac{1}{2\pi iN}\int_{\sigma-i\pi N}^{\sigma+i\pi N}e^{ns/N}\frac{F_k(e^{-s/N};N)}{1-e^{-s/N}}\,ds. \tag*{(34)} I k ( n ; N ) = 2 π i N 1 ∫ σ − iπ N σ + iπ N e n s / N 1 − e − s / N F k ( e − s / N ; N ) d s . (34) The substitution t = e − s / N t=e^{-s/N} t = e − s / N t = 1 t=1 t = 1 e − s / N = 1 − s / N + O ( N − 2 ) e^{-s/N}=1-s/N+O(N^{-2}) e − s / N = 1 − s / N + O ( N − 2 ) s s s e − s / N e^{-s/N} e − s / N 2 π i N 2\pi iN 2 π i N s s s (9) near the half-line: the corresponding change of variables puts the generating function in the scale where the cutoff parameter is visible. Therefore we study the large-N N N I ~ k ( ⌊ α N ⌋ , N ) \widetilde{I}_k(\lfloor\alpha N\rfloor,N) I k (⌊ α N ⌋ , N ) M k ( α ) M_k(\alpha) M k ( α )
M k ( α ) : = lim N → ∞ I ~ k ( ⌊ α N ⌋ , N ) N 2 k , k ∈ N , α > 0. (35) M_k(\alpha):=\lim_{N\to\infty}\frac{\widetilde{I}_k(\lfloor\alpha N\rfloor,N)}{N^{2k}},\qquad k\in\mathbb{N},\ \alpha>0. \tag*{(35)} M k ( α ) := N → ∞ lim N 2 k I k (⌊ α N ⌋ , N ) , k ∈ N , α > 0. (35) Due to (31) , we can write
I ~ k ( ⌊ α N ⌋ , N ) N 2 k = 1 N ∑ m = k ⌊ α N ⌋ I k ( m , N ) N 2 k − 1 . (36) \frac{\widetilde{I}_k(\lfloor\alpha N\rfloor,N)}{N^{2k}}=\frac{1}{N}\sum_{m=k}^{\lfloor\alpha N\rfloor}\frac{I_k(m,N)}{N^{2k-1}}. \tag*{(36)} N 2 k I k (⌊ α N ⌋ , N ) = N 1 m = k ∑ ⌊ α N ⌋ N 2 k − 1 I k ( m , N ) . (36) By [20 Proposition 5.2] , passage to the limit in this Riemann sum gives
M k ( α ) = ∫ 0 α δ k ( c ) d c . (37) M_k(\alpha)=\int_0^\alpha\delta_k(c)\,\mathrm{d}c. \tag*{(37)} M k ( α ) = ∫ 0 α δ k ( c ) d c . (37) As δ k \delta_k δ k ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) M k M_k M k
M k ′ ( α ) = δ k ( α ) , α > 0. (38) M_k'(\alpha)=\delta_k(\alpha),\qquad\alpha>0. \tag*{(38)} M k ′ ( α ) = δ k ( α ) , α > 0. (38) To determine M k ( α ) M_k(\alpha) M k ( α ) n n n N N N (28) , the total degree is n = i 1 + ⋯ + i k n=i_1+\cdots+i_k n = i 1 + ⋯ + i k N N N n ≤ N n\leq N n ≤ N [9 ] gives these evaluations for products of secular coefficients, while the Diaconis–Shahshahani formula [10 ] gives the evaluations for trace products. Beyond this range, one must retain the finite-N N N N N N n n n n > N n>N n > N n = ⌊ α N ⌋ n=\lfloor\alpha N\rfloor n = ⌊ α N ⌋ M k ( α ) M_k(\alpha) M k ( α ) α ≤ 1 \alpha\leq1 α ≤ 1 α > 1 \alpha>1 α > 1 [19 ] . Generally, one expects M k ( α ) M_k(\alpha) M k ( α ) α \alpha α
Results
We now present the main results of this paper. We begin by introducing a polynomial that appears in the leading-order moment asymptotics for both ζ ′ / ζ \zeta'/\zeta ζ ′ / ζ U ( N ) U(N) U ( N ) Theorem 2.4 .
Definition 2.1. Fix τ ∈ ( 0 , 1 2 ) \tau\in(0,\frac{1}{2}) τ ∈ ( 0 , 2 1 ) α ∈ R + \alpha\in\mathbb{R}_+ α ∈ R + 0 ≤ r ≤ k 0\leq r\leq k 0 ≤ r ≤ k
P r , k ( α ) : = 1 ( 2 π i ) 2 k ∑ m = 0 k ( − 1 ) k − m ( k m ) 1 ( k + m ) ! P_{r,k}(\alpha):=\frac{1}{(2\pi i)^{2k}}\sum_{m=0}^{k}(-1)^{k-m}\binom{k}{m}\frac{1}{(k+m)!} P r , k ( α ) := ( 2 π i ) 2 k 1 m = 0 ∑ k ( − 1 ) k − m ( m k ) ( k + m )! 1 × ∮ c j ( r ) ∮ c ^ ℓ ( r ) ( ∑ j = 1 k x j − α ) k + m Δ ( x ) 2 Δ ( y ) 2 ∏ i , j = 1 k ( x i − y j − 1 ) ∏ j = 1 k x j − k ∏ ℓ = 1 k y ℓ − k {}\times\oint_{c_j(r)}\oint_{\widehat{c}_\ell(r)}
\left(\sum_{j=1}^{k}x_j-\alpha\right)^{k+m}
\Delta(x)^2\Delta(y)^2
\prod_{i,j=1}^{k}(x_i-y_j-1)
\prod_{j=1}^{k}x_j^{-k}\prod_{\ell=1}^{k}y_\ell^{-k} × ∮ c j ( r ) ∮ c ℓ ( r ) ( j = 1 ∑ k x j − α ) k + m Δ ( x ) 2 Δ ( y ) 2 i , j = 1 ∏ k ( x i − y j − 1 ) j = 1 ∏ k x j − k ℓ = 1 ∏ k y ℓ − k × ( ∑ j = 1 k 1 x j + ∑ ℓ = 1 k 1 y ℓ + 1 ) m ( ∑ j = 1 k 1 x j − 1 + ∑ ℓ = 1 k 1 y ℓ ) k ∏ j = 1 k d x j ∏ ℓ = 1 k d y ℓ , (39) {}\times\left(\sum_{j=1}^{k}\frac{1}{x_j}+\sum_{\ell=1}^{k}\frac{1}{y_\ell+1}\right)^m
\left(\sum_{j=1}^{k}\frac{1}{x_j-1}+\sum_{\ell=1}^{k}\frac{1}{y_\ell}\right)^k
\prod_{j=1}^{k}\mathrm{d}x_j\prod_{\ell=1}^{k}\mathrm{d}y_\ell, \tag*{(39)} × ( j = 1 ∑ k x j 1 + ℓ = 1 ∑ k y ℓ + 1 1 ) m ( j = 1 ∑ k x j − 1 1 + ℓ = 1 ∑ k y ℓ 1 ) k j = 1 ∏ k d x j ℓ = 1 ∏ k d y ℓ , (39) where Δ ( u ) = ∏ i < j ( u j − u i ) \Delta(u)=\prod_{i<j}(u_j-u_i) Δ ( u ) = ∏ i < j ( u j − u i ) c j ( r ) c_j(r) c j ( r ) τ \tau τ 1 1 1 j = 1 , … , r j=1,\ldots,r j = 1 , … , r 0 0 0 c ^ ℓ ( r ) \widehat{c}_\ell(r) c ℓ ( r ) τ \tau τ − 1 -1 − 1 ℓ = 1 , … , r \ell=1,\ldots,r ℓ = 1 , … , r 0 0 0 r > k r>k r > k P r , k ( α ) : = 0 P_{r,k}(\alpha):=0 P r , k ( α ) := 0 P r , k ( α ) P_{r,k}(\alpha) P r , k ( α ) 2 r 2 2r^2 2 r 2 α = r \alpha=r α = r
The value of P r , k ( α ) P_{r,k}(\alpha) P r , k ( α ) τ ∈ ( 0 , 1 2 ) \tau\in(0,\frac{1}{2}) τ ∈ ( 0 , 2 1 ) c j ( r ) c_j(r) c j ( r ) c ^ ℓ ( r ) \widehat{c}_\ell(r) c ℓ ( r ) τ \tau τ M k ( α ) M_k(\alpha) M k ( α ) (12) in terms of P r , k ( α ) P_{r,k}(\alpha) P r , k ( α ) k ≥ 1 k\geq1 k ≥ 1 k = 1 k=1 k = 1 (15) .
Theorem 2.2. Let k ∈ N k \in\mathbb{N} k ∈ N α ∈ R + \alpha\in\mathbb{R}_{+} α ∈ R + Y ≪ T γ Y \ll T^{\gamma} Y ≪ T γ γ = max ( 1 , α / 2 ) + ε \gamma= \max(1,\alpha/2)+\varepsilon γ = max ( 1 , α /2 ) + ε ε > 0 \varepsilon> 0 ε > 0
M k ( α ) = 1 k ! 2 ∑ 0 ≤ r < α ( k r ) 2 P r , k ( α ) , \mathcal{M}_{k}(\alpha)=\frac{1}{k!^{2}}\sum_{0\le r<\alpha}\binom{k}{r}^{2}P_{r,k}(\alpha), M k ( α ) = k ! 2 1 0 ≤ r < α ∑ ( r k ) 2 P r , k ( α ) , with P r , k ( α ) P_{r,k}(\alpha) P r , k ( α ) (39) and in particular, M k ( α ) \mathcal{M}_{k}(\alpha) M k ( α ) 2 ℓ 2 − 1 2\ell^{2}-1 2 ℓ 2 − 1 α = ℓ \alpha=\ell α = ℓ ℓ = 1 , … , k \ell=1,\ldots,k ℓ = 1 , … , k
The polynomial range for the parameter Y Y Y ℑ ( z 1 − z 2 ) \Im(z_{1}-z_{2}) ℑ ( z 1 − z 2 ) ℑ z 2 \Im z_{2} ℑ z 2 (132) . The following theorem identifies the variance coefficient D k ( α ) D_{k}(\alpha) D k ( α ) (20) , with δ k ( α ) \delta_{k}(\alpha) δ k ( α ) Y Y Y M k ( α ) \mathcal{M}_{k}(\alpha) M k ( α ) Y Y Y Theorem 2.2 .
Theorem 2.3. Let k ∈ N k \in\mathbb{N} k ∈ N α ∈ R + \alpha\in\mathbb{R}_{+} α ∈ R + Y ≪ ( log T ) η Y \ll(\log T)^{\eta} Y ≪ ( log T ) η η > 0 \eta>0 η > 0
D k ( α ) = δ k ( α ) . D_{k}(\alpha)=\delta_{k}(\alpha). D k ( α ) = δ k ( α ) . Under the stronger hypothesis of Theorem 2.2 , we have D k ( α ) = M k ′ ( α ) D_{k}(\alpha)=\mathcal{M}_{k}'(\alpha) D k ( α ) = M k ′ ( α )
We also prove an identity between Q k ( α ) Q_{k}(\alpha) Q k ( α ) (21) , and the short-interval variance δ k ( α ) \delta_{k}(\alpha) δ k ( α )
Theorem 2.4. Let k ≥ 4 k\ge4 k ≥ 4 α ∈ R + \alpha\in\mathbb{R}_{+} α ∈ R + Y ≪ q α / 2 + ε Y\ll q^{\alpha/2+\varepsilon} Y ≪ q α /2 + ε ε > 0 \varepsilon>0 ε > 0
Q k ( α ) = δ k ( α ) . Q_{k}(\alpha)=\delta_{k}(\alpha). Q k ( α ) = δ k ( α ) . A much stronger decay of the error term in Conjecture 2 would extend the result to k = 1 , 2 , 3 k=1,2,3 k = 1 , 2 , 3 (30) , we prove the following result for M k ( α ) \mathcal{M}_{k}(\alpha) M k ( α ) (35) .
Theorem 2.5. Let k ∈ N k\in\mathbb{N} k ∈ N α > 0 \alpha>0 α > 0
M k ( α ) = 1 k ! 2 ∑ 0 ≤ r < α ( k r ) 2 P r , k ( α ) , \mathcal{M}_{k}(\alpha)=\frac{1}{k!^{2}}\sum_{0\le r<\alpha}\binom{k}{r}^{2}P_{r,k}(\alpha), M k ( α ) = k ! 2 1 0 ≤ r < α ∑ ( r k ) 2 P r , k ( α ) , with P r , k ( α ) P_{r,k}(\alpha) P r , k ( α ) (39) .
As established in (38) , M k ( α ) \mathcal{M}_{k}(\alpha) M k ( α ) δ k ( α ) \delta_{k}(\alpha) δ k ( α ) Theorem 2.5 , we may replace the trace-sum matrix integral definition of δ k ( α ) \delta_{k}(\alpha) δ k ( α ) D k ( α ) D_{k}(\alpha) D k ( α ) Q k ( α ) Q_{k}(\alpha) Q k ( α )
δ k ( α ) = 1 k ! 2 ∑ 0 ≤ r < α ( k r ) 2 P r , k ′ ( α ) . (40) \delta_{k}(\alpha)=\frac{1}{k!^{2}}\sum_{0\le r<\alpha}\binom{k}{r}^{2}P_{r,k}'(\alpha). \tag*{(40)} δ k ( α ) = k ! 2 1 0 ≤ r < α ∑ ( r k ) 2 P r , k ′ ( α ) . (40) To prove both Theorem 2.2 and Theorem 2.5 , we will use the following three lemmas that provide the contour identities and degree estimates needed. Lemma 2.6 decomposes the multiple contour integral arising from Conjecture 1 according to the choice of pole centre for each variable, and shows that only terms with equal numbers of u u u v v v Theorem 2.5 . Lemma 2.8 shows that only finitely many Taylor coefficients of the holomorphic factor contribute to the residue, and provides the degree bounds needed to evaluate and estimate the resulting outer contour integrals.
Lemma 2.6. Fix k ∈ Z + k \in\mathbb{Z}_{+} k ∈ Z + L ∈ C L \in\mathbb{C} L ∈ C σ ∈ C ∖ { 0 } \sigma\in\mathbb{C}\setminus\{0\} σ ∈ C ∖ { 0 } 0 < τ < 1 2 0<\tau<\frac{1}{2} 0 < τ < 2 1 μ > ( 1 + τ ) ∣ σ ∣ \mu>(1+\tau)|\sigma| μ > ( 1 + τ ) ∣ σ ∣ Φ L \Phi_L Φ L { u ∈ C k : ∣ u j ∣ ≤ μ , 1 ≤ j ≤ k } \{u\in\mathbb{C}^k: |u_j|\leq\mu,\ 1\leq j\leq k\} { u ∈ C k : ∣ u j ∣ ≤ μ , 1 ≤ j ≤ k } u 1 , … , u k u_1,\ldots,u_k u 1 , … , u k
A ( u , v ; σ ) : = ∑ j = 1 k 1 u j + ∑ ℓ = 1 k 1 v ℓ + σ − L , B ( u , v ; σ ) : = ∑ j = 1 k 1 u j − σ + ∑ ℓ = 1 k 1 v ℓ . A(u,v;\sigma):=\sum_{j=1}^{k}\frac{1}{u_j}+\sum_{\ell=1}^{k}\frac{1}{v_\ell+\sigma}-L,\qquad B(u,v;\sigma):=\sum_{j=1}^{k}\frac{1}{u_j-\sigma}+\sum_{\ell=1}^{k}\frac{1}{v_\ell}. A ( u , v ; σ ) := j = 1 ∑ k u j 1 + ℓ = 1 ∑ k v ℓ + σ 1 − L , B ( u , v ; σ ) := j = 1 ∑ k u j − σ 1 + ℓ = 1 ∑ k v ℓ 1 . Then we have
∮ ∣ u j ∣ = μ ∣ v ℓ ∣ = μ Φ L ( u 1 , … , u k ) Δ ( u ) 2 Δ ( v ) 2 ∏ j , ℓ = 1 k ( u j − v ℓ − σ ) A ( u , v ; σ ) k B ( u , v ; σ ) k ∏ j = 1 k d u j u j k ∏ ℓ = 1 k d v ℓ v ℓ k = ∑ r = 0 k ( k r ) 2 ∑ a = 0 k ( − 1 ) k − a ( k a ) L k − a σ k 2 − ( a + k ) J L , r ( a ) ( σ ) . \begin{aligned}
\oint_{\substack{|u_j|=\mu\\|v_\ell|=\mu}}\Phi_L(u_1,\ldots,u_k)\Delta(u)^2\Delta(v)^2\prod_{j,\ell=1}^{k}(u_j-v_\ell-\sigma)A(u,v;\sigma)^kB(u,v;\sigma)^k\prod_{j=1}^{k}\frac{du_j}{u_j^k}\prod_{\ell=1}^{k}\frac{dv_\ell}{v_\ell^k}\\
=\sum_{r=0}^{k}\binom{k}{r}^{2}\sum_{a=0}^{k}(-1)^{k-a}\binom{k}{a}L^{k-a}\sigma^{k^2-(a+k)}J_{L,r}^{(a)}(\sigma).
\end{aligned} ∮ ∣ u j ∣ = μ ∣ v ℓ ∣ = μ Φ L ( u 1 , … , u k ) Δ ( u ) 2 Δ ( v ) 2 j , ℓ = 1 ∏ k ( u j − v ℓ − σ ) A ( u , v ; σ ) k B ( u , v ; σ ) k j = 1 ∏ k u j k d u j ℓ = 1 ∏ k v ℓ k d v ℓ = r = 0 ∑ k ( r k ) 2 a = 0 ∑ k ( − 1 ) k − a ( a k ) L k − a σ k 2 − ( a + k ) J L , r ( a ) ( σ ) . with
J L , r ( a ) ( σ ) = ∮ c j ( r ) ∮ c ^ ℓ ( r ) Φ L ( σ x 1 , … , σ x k ) Δ ( x ) 2 Δ ( y ) 2 ∏ j , ℓ = 1 k ( x j − y ℓ − 1 ) × ( ∑ j = 1 k 1 x j + ∑ ℓ = 1 k 1 y ℓ + 1 ) a ( ∑ j = 1 k 1 x j − 1 + ∑ ℓ = 1 k 1 y ℓ ) k ∏ j = 1 k x j − k d x j ∏ ℓ = 1 k y ℓ − k d y ℓ , \begin{aligned}
J_{L,r}^{(a)}(\sigma)=\oint_{c_j(r)}\oint_{\widehat c_\ell(r)}\Phi_L(\sigma x_1,\ldots,\sigma x_k)\Delta(x)^2\Delta(y)^2\prod_{j,\ell=1}^{k}(x_j-y_\ell-1)\\
{}\times\left(\sum_{j=1}^{k}\frac{1}{x_j}+\sum_{\ell=1}^{k}\frac{1}{y_\ell+1}\right)^a\left(\sum_{j=1}^{k}\frac{1}{x_j-1}+\sum_{\ell=1}^{k}\frac{1}{y_\ell}\right)^k\prod_{j=1}^{k}x_j^{-k}dx_j\prod_{\ell=1}^{k}y_\ell^{-k}dy_\ell,
\end{aligned} J L , r ( a ) ( σ ) = ∮ c j ( r ) ∮ c ℓ ( r ) Φ L ( σ x 1 , … , σ x k ) Δ ( x ) 2 Δ ( y ) 2 j , ℓ = 1 ∏ k ( x j − y ℓ − 1 ) × ( j = 1 ∑ k x j 1 + ℓ = 1 ∑ k y ℓ + 1 1 ) a ( j = 1 ∑ k x j − 1 1 + ℓ = 1 ∑ k y ℓ 1 ) k j = 1 ∏ k x j − k d x j ℓ = 1 ∏ k y ℓ − k d y ℓ , where c j ( r ) c_j(r) c j ( r ) τ \tau τ 1 1 1 j = 1 , … , r j=1,\ldots,r j = 1 , … , r 0 0 0 c ^ ℓ ( r ) \widehat c_\ell(r) c ℓ ( r ) τ \tau τ − 1 -1 − 1 ℓ = 1 , … , r \ell=1,\ldots,r ℓ = 1 , … , r 0 0 0
Lemma 2.7. Let k ≥ 1 k\geq1 k ≥ 1 z ∈ C ∖ { 0 } z\in\mathbb{C}\setminus\{0\} z ∈ C ∖ { 0 } U = ∑ j = 1 k u j U=\sum_{j=1}^{k}u_j U = ∑ j = 1 k u j V = ∑ ℓ = 1 k v ℓ V=\sum_{\ell=1}^{k}v_\ell V = ∑ ℓ = 1 k v ℓ
A z ( u , v ) = ∑ j = 1 k 1 u j + ∑ ℓ = 1 k 1 v ℓ − z , B z ( u , v ) = ∑ j = 1 k 1 u j + z + ∑ ℓ = 1 k 1 v ℓ . A_z(u,v)=\sum_{j=1}^{k}\frac{1}{u_j}+\sum_{\ell=1}^{k}\frac{1}{v_\ell-z},\qquad B_z(u,v)=\sum_{j=1}^{k}\frac{1}{u_j+z}+\sum_{\ell=1}^{k}\frac{1}{v_\ell}. A z ( u , v ) = j = 1 ∑ k u j 1 + ℓ = 1 ∑ k v ℓ − z 1 , B z ( u , v ) = j = 1 ∑ k u j + z 1 + ℓ = 1 ∑ k v ℓ 1 . For finite sets P , Q ⊂ C P,Q\subset\mathbb{C} P , Q ⊂ C
R P , Q [ F ] : = 1 ( 2 π i ) 2 k ∫ Γ P k ∫ Γ Q k F ( u , v ) ∏ j = 1 k d u j ∏ ℓ = 1 k d v ℓ , R_{P,Q}[F]:=\frac{1}{(2\pi i)^{2k}}\int_{\Gamma_P^k}\int_{\Gamma_Q^k}F(u,v)\prod_{j=1}^{k}du_j\prod_{\ell=1}^{k}dv_\ell, R P , Q [ F ] := ( 2 π i ) 2 k 1 ∫ Γ P k ∫ Γ Q k F ( u , v ) j = 1 ∏ k d u j ℓ = 1 ∏ k d v ℓ , where Γ P \Gamma_P Γ P Γ Q \Gamma_Q Γ Q P P P Q Q Q F F F s , t ∈ C s,t\in\mathbb{C} s , t ∈ C
R { 0 , − z } , { 0 , z } [ e s U + t V W z ( u , v ) ( A z ( u , v ) − s ) k ( B z ( u , v ) − t ) k ] R_{\{0,-z\},\{0,z\}}\left[e^{sU+tV}W_z(u,v)\left(A_z(u,v)-s\right)^k\left(B_z(u,v)-t\right)^k\right] R { 0 , − z } , { 0 , z } [ e s U + t V W z ( u , v ) ( A z ( u , v ) − s ) k ( B z ( u , v ) − t ) k ] = R { 0 , − z } , { 0 , z } [ e ( s − t ) U W z ( u , v ) ( A z ( u , v ) − ( s − t ) ) k B z ( u , v ) k ] , =R_{\{0,-z\},\{0,z\}}\left[e^{(s-t)U}W_z(u,v)\left(A_z(u,v)-(s-t)\right)^kB_z(u,v)^k\right], = R { 0 , − z } , { 0 , z } [ e ( s − t ) U W z ( u , v ) ( A z ( u , v ) − ( s − t ) ) k B z ( u , v ) k ] , where
W z ( u , v ) = Δ ( u ) 2 Δ ( v ) 2 ∏ j , ℓ = 1 k ( u j − v ℓ + z ) ∏ j = 1 k u j − k ∏ ℓ = 1 k v ℓ − k . W_z(u,v)=\Delta(u)^2\Delta(v)^2\prod_{j,\ell=1}^{k}(u_j-v_\ell+z)\prod_{j=1}^{k}u_j^{-k}\prod_{\ell=1}^{k}v_\ell^{-k}. W z ( u , v ) = Δ ( u ) 2 Δ ( v ) 2 j , ℓ = 1 ∏ k ( u j − v ℓ + z ) j = 1 ∏ k u j − k ℓ = 1 ∏ k v ℓ − k . Lemma 2.8. Let J L , r ( a ) ( σ ) J_{L,r}^{(a)}(\sigma) J L , r ( a ) ( σ ) Lemma 2.6 . Expand Φ L ( σ x 1 , … , σ x k ) \Phi_L(\sigma x_1,\ldots,\sigma x_k) Φ L ( σ x 1 , … , σ x k ) x x x
Φ L ( σ x 1 , … , σ x k ) = ∑ β ∈ Z ≥ 0 r γ ∈ Z ≥ 0 k − r ϕ β , γ ( r ) ( L , σ ) ∏ j = 1 r ( x j − 1 ) β j ∏ j = r + 1 k x j γ j − r . (41) \Phi_L(\sigma x_1,\ldots,\sigma x_k)=\sum_{\substack{\beta\in\mathbb{Z}_{\geq0}^{r}\\\gamma\in\mathbb{Z}_{\geq0}^{k-r}}}\phi_{\beta,\gamma}^{(r)}(L,\sigma)\prod_{j=1}^{r}(x_j-1)^{\beta_j}\prod_{j=r+1}^{k}x_j^{\gamma_{j-r}}. \tag*{(41)} Φ L ( σ x 1 , … , σ x k ) = β ∈ Z ≥ 0 r γ ∈ Z ≥ 0 k − r ∑ ϕ β , γ ( r ) ( L , σ ) j = 1 ∏ r ( x j − 1 ) β j j = r + 1 ∏ k x j γ j − r . (41) If a < r a<r a < r J L , r ( a ) ( σ ) = 0 J_{L,r}^{(a)}(\sigma)=0 J L , r ( a ) ( σ ) = 0 a ≥ r a\geq r a ≥ r J L , r ( a ) ( σ ) J_{L,r}^{(a)}(\sigma) J L , r ( a ) ( σ )
∣ β ∣ ≤ k − r , ∣ γ ∣ ≤ a − r . |\beta|\leq k-r,\qquad|\gamma|\leq a-r. ∣ β ∣ ≤ k − r , ∣ γ ∣ ≤ a − r . Here the empty multi-index convention is used for β \beta β γ \gamma γ r = 0 r=0 r = 0 r = k r=k r = k (41) is interpreted as 1. The two propositions that follow supply additional estimates and regularity properties. Under Conjecture 1, Proposition 2.9 gives the shifted-moment decay estimate used to control the tails in the short-interval variance argument.
Proposition 2.9. Assume Conjecture 1. Under the hypotheses of Conjecture 1, with 0 < ξ < 1 0<\xi<1 0 < ξ < 1 k ≥ 3 k\ge3 k ≥ 3 0 < ξ < c 0<\xi<c 0 < ξ < c k = 1 , 2 k=1,2 k = 1 , 2
1 T ∫ 0 T ( ζ ′ ζ ( 1 2 + i t + z 1 ) ) k ( ζ ′ ζ ( 1 2 − i t − z 2 ) ) k d t ≪ k , c , ξ ( log T ) 2 k ( 1 + min ( ∣ z ∣ , 1 ) log T ) − 2 ( 1 + 1 ∣ z ∣ log T ) 2 k (42) \frac{1}{T}\int_{0}^{T}\left(\frac{\zeta'}{\zeta}\left(\frac{1}{2}+it+z_{1}\right)\right)^{k}\left(\frac{\zeta'}{\zeta}\left(\frac{1}{2}-it-z_{2}\right)\right)^{k}\,\mathrm{d}t
\ll_{k,c,\xi}(\log T)^{2k}\left(1+\min(|z|,1)\log T\right)^{-2}\left(1+\frac{1}{|z|\log T}\right)^{2k} \tag*{(42)} T 1 ∫ 0 T ( ζ ζ ′ ( 2 1 + i t + z 1 ) ) k ( ζ ζ ′ ( 2 1 − i t − z 2 ) ) k d t ≪ k , c , ξ ( log T ) 2 k ( 1 + min ( ∣ z ∣ , 1 ) log T ) − 2 ( 1 + ∣ z ∣ log T 1 ) 2 k (42) where z = z 1 − z 2 z=z_{1}-z_{2} z = z 1 − z 2 In particular,
1 T ∫ 0 T ( ζ ′ ζ ( 1 2 + i t + z 1 ) ) k ( ζ ′ ζ ( 1 2 − i t − z 2 ) ) k d t ≪ ( log T ) 2 k ( 1 + min ( ∣ z ∣ , 1 ) log T ) − 2 (43) \frac{1}{T}\int_{0}^{T}\left(\frac{\zeta'}{\zeta}\left(\frac{1}{2}+it+z_{1}\right)\right)^{k}\left(\frac{\zeta'}{\zeta}\left(\frac{1}{2}-it-z_{2}\right)\right)^{k}\,\mathrm{d}t
\ll(\log T)^{2k}\left(1+\min(|z|,1)\log T\right)^{-2} \tag*{(43)} T 1 ∫ 0 T ( ζ ζ ′ ( 2 1 + i t + z 1 ) ) k ( ζ ζ ′ ( 2 1 − i t − z 2 ) ) k d t ≪ ( log T ) 2 k ( 1 + min ( ∣ z ∣ , 1 ) log T ) − 2 (43) uniformly on any subrange on which ∣ z ∣ log T ≥ c 0 > 0 |z|\log T\ge c_{0}>0 ∣ z ∣ log T ≥ c 0 > 0 k k k c c c ξ \xi ξ c 0 c_{0} c 0
Throughout, the notation A ≪ B A\ll B A ≪ B ∣ A ∣ ≤ C ∣ B ∣ |A|\le C|B| ∣ A ∣ ≤ C ∣ B ∣ C > 0 C>0 C > 0 P r , k ( α ) P_{r,k}(\alpha) P r , k ( α ) α = r \alpha=r α = r r r r M k ( α ) M_k(\alpha) M k ( α ) α = r \alpha=r α = r
Proposition 2.10. For an integer 0 ≤ r ≤ k 0\le r\le k 0 ≤ r ≤ k P r , k ( α ) P_{r,k}(\alpha) P r , k ( α ) 2 r 2 2r^{2} 2 r 2 α = r \alpha=r α = r
Proof of results
Proof of Lemma 2.6
Fix 0 < τ < 1 / 2 0<\tau<1/2 0 < τ < 1/2 σ ≠ 0 \sigma\ne0 σ = 0 μ > ( 1 + τ ) ∣ σ ∣ \mu>(1+\tau)|\sigma| μ > ( 1 + τ ) ∣ σ ∣ μ > ∣ σ ∣ \mu>|\sigma| μ > ∣ σ ∣ ∣ u j ∣ = μ |u_j|=\mu ∣ u j ∣ = μ ∣ v ℓ ∣ = μ |v_\ell|=\mu ∣ v ℓ ∣ = μ u j = 0 , σ u_j=0,\sigma u j = 0 , σ v ℓ = 0 , − σ v_\ell=0,-\sigma v ℓ = 0 , − σ Φ L ( u ) \Phi_L(u) Φ L ( u ) { ∣ u j ∣ ≤ μ , 1 ≤ j ≤ k } \{|u_j|\le\mu,\ 1\le j\le k\} { ∣ u j ∣ ≤ μ , 1 ≤ j ≤ k } 2 k 2k 2 k
G L ( σ ) : = ∮ ∣ u j ∣ = μ ∮ ∣ v ℓ ∣ = μ Φ L ( u ) Δ ( u ) 2 Δ ( v ) 2 ∏ j , ℓ = 1 k ( u j − v ℓ − σ ) ( ∑ j = 1 k 1 u j + ∑ ℓ = 1 k 1 v ℓ + σ − L ) k ( ∑ j = 1 k 1 u j − σ + ∑ ℓ = 1 k 1 v ℓ ) k ∏ j = 1 k u j − k ∏ ℓ = 1 k v ℓ − k ∏ j = 1 k d u j ∏ ℓ = 1 k d v ℓ . (44) G_L(\sigma):=\oint_{|u_j|=\mu}\oint_{|v_\ell|=\mu}\Phi_L(u)\Delta(u)^2\Delta(v)^2\prod_{j,\ell=1}^{k}(u_j-v_\ell-\sigma)\left(\sum_{j=1}^{k}\frac{1}{u_j}+\sum_{\ell=1}^{k}\frac{1}{v_\ell+\sigma}-L\right)^k\left(\sum_{j=1}^{k}\frac{1}{u_j-\sigma}+\sum_{\ell=1}^{k}\frac{1}{v_\ell}\right)^k\prod_{j=1}^{k}u_j^{-k}\prod_{\ell=1}^{k}v_\ell^{-k}\prod_{j=1}^{k}\mathrm{d}u_j\prod_{\ell=1}^{k}\mathrm{d}v_\ell. \tag*{(44)} G L ( σ ) := ∮ ∣ u j ∣ = μ ∮ ∣ v ℓ ∣ = μ Φ L ( u ) Δ ( u ) 2 Δ ( v ) 2 j , ℓ = 1 ∏ k ( u j − v ℓ − σ ) ( j = 1 ∑ k u j 1 + ℓ = 1 ∑ k v ℓ + σ 1 − L ) k ( j = 1 ∑ k u j − σ 1 + ℓ = 1 ∑ k v ℓ 1 ) k j = 1 ∏ k u j − k ℓ = 1 ∏ k v ℓ − k j = 1 ∏ k d u j ℓ = 1 ∏ k d v ℓ . (44) Let ρ : = τ ∣ σ ∣ \rho:=\tau|\sigma| ρ := τ ∣ σ ∣ ρ \rho ρ u u u σ \sigma σ v v v − σ -\sigma − σ μ > ( 1 + τ ) ∣ σ ∣ \mu>(1+\tau)|\sigma| μ > ( 1 + τ ) ∣ σ ∣ 2 ρ = 2 τ ∣ σ ∣ < ∣ σ ∣ 2\rho=2\tau|\sigma|<|\sigma| 2 ρ = 2 τ ∣ σ ∣ < ∣ σ ∣ Φ L ( u ) \Phi_L(u) Φ L ( u )
G L ( σ ) = ∑ r = 0 k ∑ s = 0 k ( k r ) ( k s ) G L , r , s ( σ ) (45) G_L(\sigma)=\sum_{r=0}^{k}\sum_{s=0}^{k}\binom{k}{r}\binom{k}{s}G_{L,r,s}(\sigma) \tag*{(45)} G L ( σ ) = r = 0 ∑ k s = 0 ∑ k ( r k ) ( s k ) G L , r , s ( σ ) (45) where r r r u u u σ \sigma σ s s s v v v − σ -\sigma − σ ( k − r ) (k-r) ( k − r ) u u u ( k − s ) (k-s) ( k − s ) v v v Φ L \Phi_L Φ L u u u
G L , r , s ( σ ) = ∫ C j σ ( r ) ∫ C ^ ℓ σ ( s ) Φ L ( u ) ∏ j , ℓ = 1 k ( u j − v ℓ − σ ) ∏ i ≠ j ( u i − u j ) ∏ i ≠ j ( v i − v j ) × A ( u , v ; σ ) k B ( u , v ; σ ) k ∏ j u j − k ∏ ℓ v ℓ − k ∏ j = 1 k d u j ∏ ℓ = 1 k d v ℓ , (46) G_{L,r,s}(\sigma)=\int_{C_j^\sigma(r)}\int_{\widehat{C}_\ell^\sigma(s)}\Phi_L(u)\prod_{j,\ell=1}^{k}(u_j-v_\ell-\sigma)\prod_{i\ne j}(u_i-u_j)\prod_{i\ne j}(v_i-v_j)
\times A(u,v;\sigma)^k B(u,v;\sigma)^k\prod_j u_j^{-k}\prod_\ell v_\ell^{-k}\prod_{j=1}^{k}du_j\prod_{\ell=1}^{k}dv_\ell, \tag*{(46)} G L , r , s ( σ ) = ∫ C j σ ( r ) ∫ C ℓ σ ( s ) Φ L ( u ) j , ℓ = 1 ∏ k ( u j − v ℓ − σ ) i = j ∏ ( u i − u j ) i = j ∏ ( v i − v j ) × A ( u , v ; σ ) k B ( u , v ; σ ) k j ∏ u j − k ℓ ∏ v ℓ − k j = 1 ∏ k d u j ℓ = 1 ∏ k d v ℓ , (46) with
A ( u , v ; σ ) = ∑ j = 1 k 1 u j + ∑ ℓ = 1 k 1 v ℓ + σ − L , B ( u , v ; σ ) = ∑ ℓ = 1 k 1 v ℓ + ∑ j = 1 k 1 u j − σ , (47) A(u,v;\sigma)=\sum_{j=1}^{k}\frac{1}{u_j}+\sum_{\ell=1}^{k}\frac{1}{v_\ell+\sigma}-L,\qquad B(u,v;\sigma)=\sum_{\ell=1}^{k}\frac{1}{v_\ell}+\sum_{j=1}^{k}\frac{1}{u_j-\sigma}, \tag*{(47)} A ( u , v ; σ ) = j = 1 ∑ k u j 1 + ℓ = 1 ∑ k v ℓ + σ 1 − L , B ( u , v ; σ ) = ℓ = 1 ∑ k v ℓ 1 + j = 1 ∑ k u j − σ 1 , (47) for u = ( u 1 , … , u k ) u=(u_1,\ldots,u_k) u = ( u 1 , … , u k ) v = ( v 1 , … , v k ) v=(v_1,\ldots,v_k) v = ( v 1 , … , v k ) C j σ ( r ) C_j^\sigma(r) C j σ ( r ) ρ \rho ρ σ \sigma σ 1 ≤ j ≤ r 1\le j\le r 1 ≤ j ≤ r 0 0 0 r < j ≤ k r<j\le k r < j ≤ k C ^ ℓ σ ( s ) \widehat{C}_\ell^\sigma(s) C ℓ σ ( s ) ρ \rho ρ − σ -\sigma − σ 1 ≤ ℓ ≤ s 1\le\ell\le s 1 ≤ ℓ ≤ s 0 0 0 s < ℓ ≤ k s<\ell\le k s < ℓ ≤ k σ \sigma σ
A ( u , v ; σ ) k = ∑ a = 0 k ( − 1 ) k − a ( k a ) L k − a ( ∑ j = 1 k 1 u j + ∑ ℓ = 1 k 1 v ℓ + σ ) a . (48) A(u,v;\sigma)^k=\sum_{a=0}^{k}(-1)^{k-a}\binom{k}{a}L^{k-a}\left(\sum_{j=1}^{k}\frac{1}{u_j}+\sum_{\ell=1}^{k}\frac{1}{v_\ell+\sigma}\right)^a. \tag*{(48)} A ( u , v ; σ ) k = a = 0 ∑ k ( − 1 ) k − a ( a k ) L k − a ( j = 1 ∑ k u j 1 + ℓ = 1 ∑ k v ℓ + σ 1 ) a . (48) We perform a change of variables, u i ↦ σ u i u_i\mapsto\sigma u_i u i ↦ σ u i v ℓ ↦ σ v ℓ v_\ell\mapsto\sigma v_\ell v ℓ ↦ σ v ℓ σ \sigma σ (46) . For the factor ∏ i ≠ j ( u i − u j ) \prod_{i\ne j}(u_i-u_j) ∏ i = j ( u i − u j ) v v v σ 2 k ( k − 1 ) \sigma^{2k(k-1)} σ 2 k ( k − 1 ) σ k 2 \sigma^{k^2} σ k 2 k k k σ − 2 k 2 \sigma^{-2k^2} σ − 2 k 2 A ( u , v ; σ ) A(u,v;\sigma) A ( u , v ; σ ) B ( u , v ; σ ) B(u,v;\sigma) B ( u , v ; σ )
σ − ( a + k ) + k 2 . (49) \sigma^{-(a+k)+k^2}. \tag*{(49)} σ − ( a + k ) + k 2 . (49) After rescaling, the contours have radius τ \tau τ c j ( r ) c_j(r) c j ( r ) 1 1 1 j = 1 , … , r j=1,\ldots,r j = 1 , … , r 0 0 0 c ^ ℓ ( s ) \widehat{c}_\ell(s) c ℓ ( s ) − 1 -1 − 1 ℓ = 1 , … , s \ell=1,\ldots,s ℓ = 1 , … , s 0 0 0
G L , r , s ( σ ) = ∑ a = 0 k ( − 1 ) k − a ( k a ) L k − a σ k 2 − ( a + k ) J L , r , s ( a ) ( σ ) (50) G_{L,r,s}(\sigma)=\sum_{a=0}^{k}(-1)^{k-a}\binom{k}{a}L^{k-a}\sigma^{k^2-(a+k)}J_{L,r,s}^{(a)}(\sigma) \tag*{(50)} G L , r , s ( σ ) = a = 0 ∑ k ( − 1 ) k − a ( a k ) L k − a σ k 2 − ( a + k ) J L , r , s ( a ) ( σ ) (50) where
J L , r , s ( a ) ( σ ) = ∫ c j ( r ) ∫ c ^ ℓ ( s ) Φ L ( σ u ) × ∏ j , ℓ ( u j − v ℓ − 1 ) ∏ i ≠ j ( u i − u j ) ∏ i ≠ j ( v i − v j ) ∏ j u j − k ∏ ℓ v ℓ − k × ( ∑ j = 1 k 1 u j + ∑ ℓ = 1 k 1 v ℓ + 1 ) a ( ∑ ℓ = 1 k 1 v ℓ + ∑ j = 1 k 1 u j − 1 ) k ∏ j = 1 k d u j ∏ ℓ = 1 k d v ℓ . (51) \begin{aligned}
J_{L,r,s}^{(a)}(\sigma)=\int_{c_j(r)}\int_{\widehat{c}_\ell(s)}\Phi_L(\sigma u) \\
&\times\prod_{j,\ell}(u_j-v_\ell-1)\prod_{i\ne j}(u_i-u_j)\prod_{i\ne j}(v_i-v_j)\prod_j u_j^{-k}\prod_\ell v_\ell^{-k} \\
&\times\left(\sum_{j=1}^{k}\frac{1}{u_j}+\sum_{\ell=1}^{k}\frac{1}{v_\ell+1}\right)^a
\left(\sum_{\ell=1}^{k}\frac{1}{v_\ell}+\sum_{j=1}^{k}\frac{1}{u_j-1}\right)^k
\prod_{j=1}^{k}du_j\prod_{\ell=1}^{k}dv_\ell. \tag*{(51)}
\end{aligned} J L , r , s ( a ) ( σ ) = ∫ c j ( r ) ∫ c ℓ ( s ) Φ L ( σ u ) × j , ℓ ∏ ( u j − v ℓ − 1 ) i = j ∏ ( u i − u j ) i = j ∏ ( v i − v j ) j ∏ u j − k ℓ ∏ v ℓ − k × ( j = 1 ∑ k u j 1 + ℓ = 1 ∑ k v ℓ + 1 1 ) a ( ℓ = 1 ∑ k v ℓ 1 + j = 1 ∑ k u j − 1 1 ) k j = 1 ∏ k d u j ℓ = 1 ∏ k d v ℓ . (51) We now show that the contributions with r ≠ s r\ne s r = s r > s r>s r > s 0 ≤ a ≤ k 0\le a\le k 0 ≤ a ≤ k 0 < τ < 1 / 2 0<\tau<1/2 0 < τ < 1/2 0 , ± 1 0,\pm1 0 , ± 1
S ( u , v ) : = ∑ j = 1 k 1 u j + ∑ ℓ = 1 k 1 v ℓ + 1 , T ( u , v ) : = ∑ ℓ = 1 k 1 v ℓ + ∑ j = 1 k 1 u j − 1 . (52) S(u,v):=\sum_{j=1}^{k}\frac{1}{u_j}+\sum_{\ell=1}^{k}\frac{1}{v_\ell+1},\qquad T(u,v):=\sum_{\ell=1}^{k}\frac{1}{v_\ell}+\sum_{j=1}^{k}\frac{1}{u_j-1}. \tag*{(52)} S ( u , v ) := j = 1 ∑ k u j 1 + ℓ = 1 ∑ k v ℓ + 1 1 , T ( u , v ) := ℓ = 1 ∑ k v ℓ 1 + j = 1 ∑ k u j − 1 1 . (52) Thus the two bracketed terms in J L , r , s ( a ) ( σ ) J_{L,r,s}^{(a)}(\sigma) J L , r , s ( a ) ( σ ) S ( u , v ) a S(u,v)^a S ( u , v ) a T ( u , v ) k T(u,v)^k T ( u , v ) k
Shifted pole-count. For j ≤ r j \le r j ≤ r u j = 1 u_j = 1 u j = 1 T ( u , v ) k T(u,v)^k T ( u , v ) k ℓ ≤ s \ell\le s ℓ ≤ s v ℓ = − 1 v_\ell= -1 v ℓ = − 1 S ( u , v ) a S(u,v)^a S ( u , v ) a s > a s > a s > a 0 0 0 S ( u , v ) a S(u,v)^a S ( u , v ) a v 1 , … , v s v_1,\ldots,v_s v 1 , … , v s s ≤ a s \le a s ≤ a u 1 , … , u r u_1,\ldots,u_r u 1 , … , u r u u u 1 1 1 v s + 1 , … , v k v_{s+1},\ldots,v_k v s + 1 , … , v k v v v 0 0 0
Fix a monomial term in the multinomial expansion of T ( u , v ) k T(u,v)^k T ( u , v ) k u u u v v v
( u 1 − 1 ) − α 1 ⋯ ( u r − 1 ) − α r v s + 1 − β s + 1 ⋯ v k − β k × (other factors) , (53) (u_1-1)^{-\alpha_1}\cdots(u_r-1)^{-\alpha_r}v_{s+1}^{-\beta_{s+1}}\cdots v_k^{-\beta_k}\times\text{(other factors)}, \tag*{(53)} ( u 1 − 1 ) − α 1 ⋯ ( u r − 1 ) − α r v s + 1 − β s + 1 ⋯ v k − β k × (other factors) , (53) where α j ≥ 0 \alpha_j \ge0 α j ≥ 0 ( u j − 1 ) − 1 (u_j-1)^{-1} ( u j − 1 ) − 1 β ℓ ≥ 0 \beta_\ell\ge0 β ℓ ≥ 0 v ℓ − 1 v_\ell^{-1} v ℓ − 1 T ( u , v ) k T(u,v)^k T ( u , v ) k u j = 1 u_j=1 u j = 1
α j ≥ 1 ( 1 ≤ j ≤ r ) , (54) \alpha_j \ge1 \qquad(1 \le j \le r), \tag*{(54)} α j ≥ 1 ( 1 ≤ j ≤ r ) , (54) since otherwise the iterated residue across u 1 , … , u r u_1,\ldots,u_r u 1 , … , u r
B : = ∑ ℓ = s + 1 k β ℓ , E : = ∑ j = 1 r ( α j − 1 ) ≥ 0 , so that ∑ j = 1 r α j = r + E . (55) B := \sum_{\ell=s+1}^{k}\beta_\ell,\qquad E := \sum_{j=1}^{r}(\alpha_j-1) \ge0,\qquad\text{so that}\qquad\sum_{j=1}^{r}\alpha_j=r+E. \tag*{(55)} B := ℓ = s + 1 ∑ k β ℓ , E := j = 1 ∑ r ( α j − 1 ) ≥ 0 , so that j = 1 ∑ r α j = r + E . (55) Since the chosen monomial in T ( u , v ) k T(u,v)^k T ( u , v ) k k k k r + E r+E r + E
( u 1 − 1 ) − α 1 ⋯ ( u r − 1 ) − α r , (56) (u_1-1)^{-\alpha_1}\cdots(u_r-1)^{-\alpha_r}, \tag*{(56)} ( u 1 − 1 ) − α 1 ⋯ ( u r − 1 ) − α r , (56) we have the upper bound
B ≤ k − r − E . (57) B \le k-r-E. \tag*{(57)} B ≤ k − r − E . (57) Next we obtain a lower bound for B B B v v v v s + 1 , … , v k v_{s+1},\ldots,v_k v s + 1 , … , v k v s + 1 , … , v k v_{s+1},\ldots,v_k v s + 1 , … , v k u u u
∏ j , ℓ = 1 k ( u j − v ℓ − 1 ) . (58) \prod_{j,\ell=1}^{k}(u_j-v_\ell-1). \tag*{(58)} j , ℓ = 1 ∏ k ( u j − v ℓ − 1 ) . (58) In particular, consider the block involving the shifted u u u v v v
∏ j = 1 r ∏ ℓ = s + 1 k ( u j − v ℓ − 1 ) = ∏ j = 1 r ∏ ℓ = s + 1 k ( ( u j − 1 ) − v ℓ ) . (59) \prod_{j=1}^{r}\prod_{\ell=s+1}^{k}(u_j-v_\ell-1)=\prod_{j=1}^{r}\prod_{\ell=s+1}^{k}\bigl((u_j-1)-v_\ell\bigr). \tag*{(59)} j = 1 ∏ r ℓ = s + 1 ∏ k ( u j − v ℓ − 1 ) = j = 1 ∏ r ℓ = s + 1 ∏ k ( ( u j − 1 ) − v ℓ ) . (59) For each fixed j ≤ r j \le r j ≤ r
C j ( u j ; v s + 1 , … , v k ) : = ∏ ℓ = s + 1 k ( ( u j − 1 ) − v ℓ ) . (60) C_j(u_j;v_{s+1},\ldots,v_k):=\prod_{\ell=s+1}^{k}\bigl((u_j-1)-v_\ell\bigr). \tag*{(60)} C j ( u j ; v s + 1 , … , v k ) := ℓ = s + 1 ∏ k ( ( u j − 1 ) − v ℓ ) . (60) Expanding C j C_j C j u j − 1 u_j-1 u j − 1
C j ( u j ; v s + 1 , … , v k ) = ∑ p = 0 k − s ( u j − 1 ) p Q p ( v s + 1 , … , v k ) , (61) C_j(u_j;v_{s+1},\ldots,v_k)=\sum_{p=0}^{k-s}(u_j-1)^pQ_p(v_{s+1},\ldots,v_k), \tag*{(61)} C j ( u j ; v s + 1 , … , v k ) = p = 0 ∑ k − s ( u j − 1 ) p Q p ( v s + 1 , … , v k ) , (61) where
Q p ( v s + 1 , … , v k ) = ( − 1 ) k − s − p e k − s − p ( v s + 1 , … , v k ) , (62) Q_p(v_{s+1},\ldots,v_k)=(-1)^{k-s-p}e_{k-s-p}(v_{s+1},\ldots,v_k), \tag*{(62)} Q p ( v s + 1 , … , v k ) = ( − 1 ) k − s − p e k − s − p ( v s + 1 , … , v k ) , (62) and e k − s − p e_{k-s-p} e k − s − p k − s − p k-s-p k − s − p v s + 1 , … , v k v_{s+1},\ldots,v_k v s + 1 , … , v k Q p Q_p Q p v v v
k − s − p . (63) k-s-p. \tag*{(63)} k − s − p . (63) Equivalently, choosing p p p u j − 1 u_j-1 u j − 1 k − s k-s k − s k − s − p k-s-p k − s − p v v v ( u j − 1 ) − α j (u_j-1)^{-\alpha_j} ( u j − 1 ) − α j u j = 1 u_j=1 u j = 1 ( u j − 1 ) α j − 1 (u_j-1)^{\alpha_j-1} ( u j − 1 ) α j − 1
( u j − 1 ) p Q p ( v s + 1 , … , v k ) (64) (u_j-1)^p Q_p(v_{s+1},\ldots,v_k) \tag*{(64)} ( u j − 1 ) p Q p ( v s + 1 , … , v k ) (64) contributes to the coefficient of ( u j − 1 ) α j − 1 (u_j-1)^{\alpha_j-1} ( u j − 1 ) α j − 1
p ≤ α j − 1. (65) p\leq\alpha_j-1. \tag*{(65)} p ≤ α j − 1. (65) Hence the total v v v Q p Q_p Q p v s + 1 , … , v k v_{s+1},\ldots,v_k v s + 1 , … , v k
k − s − p ≥ k − s − ( α j − 1 ) . (66) k-s-p\geq k-s-(\alpha_j-1). \tag*{(66)} k − s − p ≥ k − s − ( α j − 1 ) . (66) In other words, the j j j k − s k-s k − s v v v α j − 1 \alpha_j-1 α j − 1 α j − 1 \alpha_j-1 α j − 1 j j j v v v
k − s − ( α j − 1 ) . (67) k-s-(\alpha_j-1). \tag*{(67)} k − s − ( α j − 1 ) . (67) Since the total number of such differentiations is
∑ j = 1 r ( α j − 1 ) = E , (68) \sum_{j=1}^{r}(\alpha_j-1)=E, \tag*{(68)} j = 1 ∑ r ( α j − 1 ) = E , (68) after taking the u 1 , … , u r u_1,\ldots,u_r u 1 , … , u r v v v v s + 1 , … , v k v_{s+1},\ldots,v_k v s + 1 , … , v k
∑ j = 1 r ( k − s − ( α j − 1 ) ) = r ( k − s ) − ∑ j = 1 r ( α j − 1 ) (69) \sum_{j=1}^{r}\bigl(k-s-(\alpha_j-1)\bigr)
=r(k-s)-\sum_{j=1}^{r}(\alpha_j-1) \tag*{(69)} j = 1 ∑ r ( k − s − ( α j − 1 ) ) = r ( k − s ) − j = 1 ∑ r ( α j − 1 ) (69) = r ( k − s ) − E . (70) =r(k-s)-E. \tag*{(70)} = r ( k − s ) − E . (70) This bound may be weak if E > r ( k − s ) E>r(k-s) E > r ( k − s ) v s + 1 , … , v k v_{s+1},\ldots,v_k v s + 1 , … , v k v v v
∏ s < i ≠ j ≤ k ( v i − v j ) . (71) \prod_{s<i\ne j\leq k}(v_i-v_j). \tag*{(71)} s < i = j ≤ k ∏ ( v i − v j ) . (71) It contributes total v v v ( k − s ) ( k − s − 1 ) (k-s)(k-s-1) ( k − s ) ( k − s − 1 ) v v v v ℓ − k v_\ell^{-k} v ℓ − k v s + 1 , … , v k v_{s+1},\ldots,v_k v s + 1 , … , v k v v v u 1 = ⋯ = u r = 1 u_1=\cdots=u_r=1 u 1 = ⋯ = u r = 1 v v v
( k − s ) ( k − s − 1 ) + r ( k − s ) − E . (72) (k-s)(k-s-1)+r(k-s)-E. \tag*{(72)} ( k − s ) ( k − s − 1 ) + r ( k − s ) − E . (72) Now consider the poles at v s + 1 , … , v k = 0 v_{s+1},\ldots,v_k=0 v s + 1 , … , v k = 0 ℓ > s \ell>s ℓ > s v ℓ − k v_\ell^{-k} v ℓ − k T ( u , v ) k T(u,v)^k T ( u , v ) k v ℓ − β ℓ v_\ell^{-\beta_\ell} v ℓ − β ℓ v ℓ = 0 v_\ell=0 v ℓ = 0 k + β ℓ k+\beta_\ell k + β ℓ v ℓ = 0 v_\ell=0 v ℓ = 0 k + β ℓ − 1 k+\beta_\ell-1 k + β ℓ − 1 v ℓ v_\ell v ℓ v s + 1 = ⋯ = v k = 0 v_{s+1}=\cdots=v_k=0 v s + 1 = ⋯ = v k = 0 v v v
∑ ℓ = s + 1 k ( k + β ℓ − 1 ) = ( k − s ) ( k − 1 ) + ∑ ℓ = s + 1 k β ℓ (73) \sum_{\ell=s+1}^{k}(k+\beta_\ell-1)
=(k-s)(k-1)+\sum_{\ell=s+1}^{k}\beta_\ell \tag*{(73)} ℓ = s + 1 ∑ k ( k + β ℓ − 1 ) = ( k − s ) ( k − 1 ) + ℓ = s + 1 ∑ k β ℓ (73) = ( k − s ) ( k − 1 ) + B . (74) =(k-s)(k-1)+B. \tag*{(74)} = ( k − s ) ( k − 1 ) + B . (74) Then a necessary condition for a non-zero contribution is
( k − s ) ( k − 1 ) + B ≥ ( k − s ) ( k − s − 1 ) + r ( k − s ) − E , (75) (k-s)(k-1)+B \ge(k-s)(k-s-1)+r(k-s)-E, \tag*{(75)} ( k − s ) ( k − 1 ) + B ≥ ( k − s ) ( k − s − 1 ) + r ( k − s ) − E , (75) which rearranges to
B ≥ ( k − s ) ( r − s ) − E . (76) B \ge(k-s)(r-s)-E. \tag*{(76)} B ≥ ( k − s ) ( r − s ) − E . (76) Combining (57) and (76) gives
k − r − E ≥ ( k − s ) ( r − s ) − E , (77) k-r-E \ge(k-s)(r-s)-E, \tag*{(77)} k − r − E ≥ ( k − s ) ( r − s ) − E , (77) and hence
k − r ≥ ( k − s ) ( r − s ) . (78) k-r \ge(k-s)(r-s). \tag*{(78)} k − r ≥ ( k − s ) ( r − s ) . (78) Let d : = r − s ≥ 1 d:=r-s\ge1 d := r − s ≥ 1
( k − s ) ( r − s ) = ( k − r + d ) d = ( k − r ) d + d 2 > k − r , (79) \begin{aligned}
(k-s)(r-s) &= (k-r+d)d \\
&= (k-r)d+d^{2} \\
&> k-r, \tag*{(79)}
\end{aligned} ( k − s ) ( r − s ) = ( k − r + d ) d = ( k − r ) d + d 2 > k − r , (79) which is a contradiction. Hence the corresponding contribution is zero whenever r > s r>s r > s s > r s>r s > r u u u v v v S ( u , v ) a S(u,v)^{a} S ( u , v ) a T ( u , v ) k T(u,v)^{k} T ( u , v ) k s > r s>r s > r r ≠ s r\ne s r = s J L , r , s ( a ) ( σ ) = 0 J_{L,r,s}^{(a)}(\sigma)=0 J L , r , s ( a ) ( σ ) = 0 G L , r , s ( σ ) = 0 G_{L,r,s}(\sigma)=0 G L , r , s ( σ ) = 0 r ≠ s r\ne s r = s
G L ( σ ) = ∑ r = 0 k ( k r ) 2 ∑ a = 0 k ( − 1 ) k − a ( k a ) L k − a σ k 2 − ( a + k ) J L , r ( a ) ( σ ) , (80) G_L(\sigma)=\sum_{r=0}^{k}\binom{k}{r}^{2}\sum_{a=0}^{k}(-1)^{k-a}\binom{k}{a}L^{k-a}\sigma^{k^{2}-(a+k)}J_{L,r}^{(a)}(\sigma), \tag*{(80)} G L ( σ ) = r = 0 ∑ k ( r k ) 2 a = 0 ∑ k ( − 1 ) k − a ( a k ) L k − a σ k 2 − ( a + k ) J L , r ( a ) ( σ ) , (80) where J L , r ( a ) ( σ ) : = J L , r , r ( a ) ( σ ) J_{L,r}^{(a)}(\sigma):=J_{L,r,r}^{(a)}(\sigma) J L , r ( a ) ( σ ) := J L , r , r ( a ) ( σ ) (51) which is the claimed result.
Proof of Lemma 2.7
If P , Q ⊂ C P,Q\subset\mathbb{C} P , Q ⊂ C F ( u , v ) F(u,v) F ( u , v )
R P , Q [ F ] : = 1 ( 2 π i ) 2 k ∮ Γ P k ∮ Γ Q k F ( u , v ) ∏ j = 1 k d u j ∏ ℓ = 1 k d v ℓ , R_{P,Q}[F]:=\frac{1}{(2\pi i)^{2k}}\oint_{\Gamma_P^k}\oint_{\Gamma_Q^k}F(u,v)\prod_{j=1}^{k}du_j\prod_{\ell=1}^{k}dv_\ell, R P , Q [ F ] := ( 2 π i ) 2 k 1 ∮ Γ P k ∮ Γ Q k F ( u , v ) j = 1 ∏ k d u j ℓ = 1 ∏ k d v ℓ , where Γ P \Gamma_P Γ P Γ Q \Gamma_Q Γ Q P P P Q Q Q F F F R P , Q [ F ] R_{P,Q}[F] R P , Q [ F ] 2 k 2k 2 k u j u_j u j P P P 1 ≤ j ≤ k 1\le j\le k 1 ≤ j ≤ k v ℓ v_\ell v ℓ Q Q Q 1 ≤ ℓ ≤ k 1\le\ell\le k 1 ≤ ℓ ≤ k u j = 0 , − z u_j=0,-z u j = 0 , − z v ℓ = 0 , z v_\ell=0,z v ℓ = 0 , z
a = ( a 1 , … , a k ) , b = ( b 1 , … , b k ) , \mathbf{a}=(a_1,\ldots,a_k),\qquad\mathbf{b}=(b_1,\ldots,b_k), a = ( a 1 , … , a k ) , b = ( b 1 , … , b k ) , and, for fixed z ≠ 0 z\ne0 z = 0
X : = { a 1 , … , a k , b 1 − z , … , b k − z } , X + z : = { a 1 + z , … , a k + z , b 1 , … , b k } . \mathcal{X}:=\{a_1,\ldots,a_k,b_1-z,\ldots,b_k-z\},\qquad
\mathcal{X}+z:=\{a_1+z,\ldots,a_k+z,b_1,\ldots,b_k\}. X := { a 1 , … , a k , b 1 − z , … , b k − z } , X + z := { a 1 + z , … , a k + z , b 1 , … , b k } . We assume that the points of X \mathcal{X} X 2 k 2k 2 k
J s , t ( a , b ; z ) : = R X , X + z [ e s U + t V W z ( u , v ) × ∏ r = 1 k [ e − s a r ∏ j = 1 k ( 1 − a r u j ) − 1 ∏ ℓ = 1 k ( 1 − a r v ℓ − z ) − 1 ] × ∏ r = 1 k [ e − t b r ∏ j = 1 k ( 1 − b r u j + z ) − 1 ∏ ℓ = 1 k ( 1 − b r v ℓ ) − 1 ] ] . (81) \begin{aligned}
J_{s,t}(\mathbf{a},\mathbf{b};z)
:=R_{\mathcal{X},\mathcal{X}+z}\Bigg[
&e^{sU+tV}W_z(u,v)\\
&\times\prod_{r=1}^{k}\left[
e^{-sa_r}\prod_{j=1}^{k}\left(1-\frac{a_r}{u_j}\right)^{-1}
\prod_{\ell=1}^{k}\left(1-\frac{a_r}{v_\ell-z}\right)^{-1}
\right]\\
&\times\prod_{r=1}^{k}\left[
e^{-tb_r}\prod_{j=1}^{k}\left(1-\frac{b_r}{u_j+z}\right)^{-1}
\prod_{\ell=1}^{k}\left(1-\frac{b_r}{v_\ell}\right)^{-1}
\right]
\Bigg]. \tag*{(81)}
\end{aligned} J s , t ( a , b ; z ) := R X , X + z [ e s U + t V W z ( u , v ) × r = 1 ∏ k [ e − s a r j = 1 ∏ k ( 1 − u j a r ) − 1 ℓ = 1 ∏ k ( 1 − v ℓ − z a r ) − 1 ] × r = 1 ∏ k [ e − t b r j = 1 ∏ k ( 1 − u j + z b r ) − 1 ℓ = 1 ∏ k ( 1 − v ℓ b r ) − 1 ] ] . (81) for s , t ∈ C s,t \in\mathbb{C} s , t ∈ C
W z ( u , v ) = Δ ( u ) 2 Δ ( v ) 2 ∏ j , ℓ = 1 k ( u j − v ℓ + z ) ∏ j = 1 k u j − k ∏ ℓ = 1 k v ℓ − k . (82) W_z(u,v)=\Delta(u)^2\Delta(v)^2\prod_{j,\ell=1}^{k}(u_j-v_\ell+z)\prod_{j=1}^{k}u_j^{-k}\prod_{\ell=1}^{k}v_\ell^{-k}. \tag*{(82)} W z ( u , v ) = Δ ( u ) 2 Δ ( v ) 2 j , ℓ = 1 ∏ k ( u j − v ℓ + z ) j = 1 ∏ k u j − k ℓ = 1 ∏ k v ℓ − k . (82) Differentiating (81) once with respect to each a r a_r a r b r b_r b r a 1 = ⋯ = a k = b 1 = ⋯ = b k = 0 a_1=\cdots=a_k=b_1=\cdots=b_k=0 a 1 = ⋯ = a k = b 1 = ⋯ = b k = 0 A z ( u , v ) A_z(u,v) A z ( u , v ) B z ( u , v ) B_z(u,v) B z ( u , v )
∂ ∂ a r [ e − s a r ∏ j = 1 k ( 1 − a r u j ) − 1 ∏ ℓ = 1 k ( 1 − a r v ℓ − z ) − 1 ] ∣ a r = 0 = A z ( u , v ) − s , \left.\frac{\partial}{\partial a_r}\left[e^{-sa_r}\prod_{j=1}^{k}\left(1-\frac{a_r}{u_j}\right)^{-1}\prod_{\ell=1}^{k}\left(1-\frac{a_r}{v_\ell-z}\right)^{-1}\right]\right|_{a_r=0}=A_z(u,v)-s, ∂ a r ∂ [ e − s a r j = 1 ∏ k ( 1 − u j a r ) − 1 ℓ = 1 ∏ k ( 1 − v ℓ − z a r ) − 1 ] a r = 0 = A z ( u , v ) − s , and similarly
∂ ∂ b r [ e − t b r ∏ j = 1 k ( 1 − b r u j + z ) − 1 ∏ ℓ = 1 k ( 1 − b r v ℓ ) − 1 ] ∣ b r = 0 = B z ( u , v ) − t . \left.\frac{\partial}{\partial b_r}\left[e^{-tb_r}\prod_{j=1}^{k}\left(1-\frac{b_r}{u_j+z}\right)^{-1}\prod_{\ell=1}^{k}\left(1-\frac{b_r}{v_\ell}\right)^{-1}\right]\right|_{b_r=0}=B_z(u,v)-t. ∂ b r ∂ [ e − t b r j = 1 ∏ k ( 1 − u j + z b r ) − 1 ℓ = 1 ∏ k ( 1 − v ℓ b r ) − 1 ] b r = 0 = B z ( u , v ) − t . Hence
R { 0 , − z } , { 0 , z } [ e s U + t V W z ( u , v ) ( A z ( u , v ) − s ) k ( B z ( u , v ) − t ) k ] = ∂ ∂ a 1 ⋯ ∂ ∂ a k ∂ ∂ b 1 ⋯ ∂ ∂ b k J s , t ( a , b ; z ) ∣ a = b = 0 . (83) \mathcal{R}_{\{0,-z\},\{0,z\}}\left[e^{sU+tV}W_z(u,v)(A_z(u,v)-s)^k(B_z(u,v)-t)^k\right]
=
\left.\frac{\partial}{\partial a_1}\cdots\frac{\partial}{\partial a_k}\frac{\partial}{\partial b_1}\cdots\frac{\partial}{\partial b_k}\mathcal{J}_{s,t}(\mathbf{a},\mathbf{b};z)\right|_{\mathbf{a}=\mathbf{b}=0}. \tag*{(83)} R { 0 , − z } , { 0 , z } [ e s U + t V W z ( u , v ) ( A z ( u , v ) − s ) k ( B z ( u , v ) − t ) k ] = ∂ a 1 ∂ ⋯ ∂ a k ∂ ∂ b 1 ∂ ⋯ ∂ b k ∂ J s , t ( a , b ; z ) a = b = 0 . (83) It remains to show that J s , t \mathcal{J}_{s,t} J s , t s s s t t t s − t s-t s − t J s , t \mathcal{J}_{s,t} J s , t u j u_j u j a r a_r a r b r b_r b r
∏ r = 1 k ( 1 − a r u j ) − 1 ( 1 − b r u j + z ) − 1 = u j k ( u j + z ) k ∏ r = 1 k ( u j − a r ) ∏ r = 1 k ( u j + z − b r ) . (84) \prod_{r=1}^{k}\left(1-\frac{a_r}{u_j}\right)^{-1}\left(1-\frac{b_r}{u_j+z}\right)^{-1}
=
\frac{u_j^k(u_j+z)^k}{\prod_{r=1}^{k}(u_j-a_r)\prod_{r=1}^{k}(u_j+z-b_r)}. \tag*{(84)} r = 1 ∏ k ( 1 − u j a r ) − 1 ( 1 − u j + z b r ) − 1 = ∏ r = 1 k ( u j − a r ) ∏ r = 1 k ( u j + z − b r ) u j k ( u j + z ) k . (84) As W z W_z W z u j − k u_j^{-k} u j − k u j u_j u j u j = a r u_j=a_r u j = a r u j = b r − z u_j=b_r-z u j = b r − z 1 ≤ r ≤ k 1\le r\le k 1 ≤ r ≤ k v ℓ v_\ell v ℓ
∏ r = 1 k ( 1 − a r v ℓ − z ) − 1 ( 1 − b r v ℓ ) − 1 = ( v ℓ − z ) k v ℓ k ∏ r = 1 k ( v ℓ − z − a r ) ∏ r = 1 k ( v ℓ − b r ) . (85) \prod_{r=1}^{k}\left(1-\frac{a_r}{v_\ell-z}\right)^{-1}\left(1-\frac{b_r}{v_\ell}\right)^{-1}
=
\frac{(v_\ell-z)^kv_\ell^k}{\prod_{r=1}^{k}(v_\ell-z-a_r)\prod_{r=1}^{k}(v_\ell-b_r)}. \tag*{(85)} r = 1 ∏ k ( 1 − v ℓ − z a r ) − 1 ( 1 − v ℓ b r ) − 1 = ∏ r = 1 k ( v ℓ − z − a r ) ∏ r = 1 k ( v ℓ − b r ) ( v ℓ − z ) k v ℓ k . (85) and since W z W_z W z v ℓ − k v_\ell^{-k} v ℓ − k v ℓ v_\ell v ℓ v ℓ = a r + z v_\ell=a_r+z v ℓ = a r + z v ℓ = b r v_\ell=b_r v ℓ = b r u u u u = x u=x u = x x ∈ X x\in\mathcal{X} x ∈ X v v v v = x + z v=x+z v = x + z x ∈ X x\in\mathcal{X} x ∈ X
When expanding the 2 k 2k 2 k u j u_j u j x ∈ X x\in\mathcal{X} x ∈ X v ℓ v_\ell v ℓ x + z x+z x + z x ∈ X x\in\mathcal{X} x ∈ X x x x u u u v v v
X = { a 1 , a 2 , b 1 − z , b 2 − z } \mathcal{X}=\{a_1,a_2,b_1-z,b_2-z\} X = { a 1 , a 2 , b 1 − z , b 2 − z } when k = 2 k=2 k = 2 u u u v v v
u 1 = a 1 , u 2 = b 2 − z , v 1 = a 1 + z , v 2 = b 2 . u_1=a_1,\qquad u_2=b_2-z,\qquad v_1=a_1+z,\qquad v_2=b_2. u 1 = a 1 , u 2 = b 2 − z , v 1 = a 1 + z , v 2 = b 2 . We will now show that such a paired choice does not, in fact, contribute to the 2 k 2k 2 k u u u S ⊂ X S\subset\mathcal{X} S ⊂ X k k k v v v X ∖ S \mathcal{X}\setminus S X ∖ S z z z
Suppose first that a candidate local residue contains both a u u u x x x v v v x + z x+z x + z x ∈ X x\in\mathcal{X} x ∈ X p p p q q q p = 0 p=0 p = 0 u = x u=x u = x q = 0 q=0 q = 0 v = x + z v=x+z v = x + z
u = x + p , v = x + z + q , (86) u=x+p,\qquad v=x+z+q, \tag*{(86)} u = x + p , v = x + z + q , (86) for which the mixed factor
∏ j , ℓ = 1 k ( u j − v ℓ + z ) \prod_{j,\ell=1}^{k}(u_j-v_\ell+z) j , ℓ = 1 ∏ k ( u j − v ℓ + z ) supplies
u − v + z = p − q . u-v+z=p-q. u − v + z = p − q . The corresponding simple pole factors from (84) and (85) give 1 / ( p q ) 1/(pq) 1/ ( pq ) p p p q q q ( 0 , 0 ) (0,0) ( 0 , 0 )
p − q p q = 1 q − 1 p . \frac{p-q}{pq}=\frac{1}{q}-\frac{1}{p}. pq p − q = q 1 − p 1 . This Laurent expansion has no p − 1 q − 1 p^{-1}q^{-1} p − 1 q − 1 u u u x ∈ X x\in\mathcal{X} x ∈ X
u i = x + p , u j = x + q , u_i=x+p,\qquad u_j=x+q, u i = x + p , u j = x + q , then the local contribution from the Vandermonde and the two simple poles is
( u i − u j ) 2 ( u i − x ) ( u j − x ) = ( p − q ) 2 p q = p q − 2 + q p . \frac{(u_i-u_j)^2}{(u_i-x)(u_j-x)}
=\frac{(p-q)^2}{pq}
=\frac{p}{q}-2+\frac{q}{p}. ( u i − x ) ( u j − x ) ( u i − u j ) 2 = pq ( p − q ) 2 = q p − 2 + p q . Again, after multiplication by factors holomorphic in p p p q q q p − 1 q − 1 p^{-1}q^{-1} p − 1 q − 1 v v v x ∈ X x\in\mathcal{X} x ∈ X u u u u = x u=x u = x v v v v = x + z v=x+z v = x + z S ⊂ X S\subset\mathcal{X} S ⊂ X k k k u u u X ∖ S \mathcal{X}\setminus S X ∖ S z z z v v v
U = ∑ x ∈ S x , V = ∑ x ∈ X ∖ S ( x + z ) , U=\sum_{x\in S}x,\qquad V=\sum_{x\in\mathcal{X}\setminus S}(x+z), U = x ∈ S ∑ x , V = x ∈ X ∖ S ∑ ( x + z ) , as the u u u S S S v v v x + z x+z x + z x ∈ X ∖ S x\in\mathcal{X}\setminus S x ∈ X ∖ S
∑ x ∈ X x = ∑ r = 1 k a r + ∑ r = 1 k ( b r − z ) = ∑ r = 1 k a r + ∑ r = 1 k b r − k z , \sum_{x\in\mathcal{X}}x
=\sum_{r=1}^{k}a_r+\sum_{r=1}^{k}(b_r-z)
=\sum_{r=1}^{k}a_r+\sum_{r=1}^{k}b_r-kz, x ∈ X ∑ x = r = 1 ∑ k a r + r = 1 ∑ k ( b r − z ) = r = 1 ∑ k a r + r = 1 ∑ k b r − k z , and as ∣ X ∖ S ∣ = k |\mathcal{X}\setminus S|=k ∣ X ∖ S ∣ = k
V = ∑ x ∈ X ∖ S x + k z = ( ∑ r = 1 k a r + ∑ r = 1 k b r − k z − U ) + k z = ∑ r = 1 k a r + ∑ r = 1 k b r − U . (87) \begin{aligned}
V&=\sum_{x\in\mathcal{X}\setminus S}x+kz\\
&=\left(\sum_{r=1}^{k}a_r+\sum_{r=1}^{k}b_r-kz-U\right)+kz\\
&=\sum_{r=1}^{k}a_r+\sum_{r=1}^{k}b_r-U. \tag*{(87)}
\end{aligned} V = x ∈ X ∖ S ∑ x + k z = ( r = 1 ∑ k a r + r = 1 ∑ k b r − k z − U ) + k z = r = 1 ∑ k a r + r = 1 ∑ k b r − U . (87) Using the definition of J s , t ( a , b ; z ) J_{s,t}(\mathbf{a},\mathbf{b};z) J s , t ( a , b ; z ) (81) , consider the exponential part of each contributing local residue
e s U + t V e − s ∑ r = 1 k a r e − t ∑ r = 1 k b r . e^{sU+tV}e^{-s\sum_{r=1}^{k}a_r}e^{-t\sum_{r=1}^{k}b_r}. e s U + t V e − s ∑ r = 1 k a r e − t ∑ r = 1 k b r . Substituting in the form of V V V (87) , this becomes
e s U + t ( ∑ a r + ∑ b r − U ) e − s ∑ a r e − t ∑ b r = e ( s − t ) U e − ( s − t ) ∑ a r . e^{sU+t\left(\sum a_r+\sum b_r-U\right)}e^{-s\sum a_r}e^{-t\sum b_r}
=e^{(s-t)U}e^{-(s-t)\sum a_r}. e s U + t ( ∑ a r + ∑ b r − U ) e − s ∑ a r e − t ∑ b r = e ( s − t ) U e − ( s − t ) ∑ a r . This is exactly the exponential factor that appears in the corresponding local contribution to J s − t , 0 ( a , b ; z ) \mathcal{J}_{s-t,0}(\mathbf{a},\mathbf{b};z) J s − t , 0 ( a , b ; z ) s s s t t t J s , t \mathcal{J}_{s,t} J s , t J s − t , 0 \mathcal{J}_{s-t,0} J s − t , 0
J s , t ( a , b ; z ) = J s − t , 0 ( a , b ; z ) . \mathcal{J}_{s,t}(\mathbf{a},\mathbf{b};z)=\mathcal{J}_{s-t,0}(\mathbf{a},\mathbf{b};z). J s , t ( a , b ; z ) = J s − t , 0 ( a , b ; z ) . Finally, taking derivatives of both sides as in (83) and then setting a = b = 0 \mathbf{a}=\mathbf{b}=0 a = b = 0
R { 0 , − z } , { 0 , z } [ e s U + t V W z ( u , v ) ( A z ( u , v ) − s ) k ( B z ( u , v ) − t ) k ] = R { 0 , − z } , { 0 , z } [ e ( s − t ) U W z ( u , v ) ( A z ( u , v ) − ( s − t ) ) k B z ( u , v ) k ] . (88) \mathcal{R}_{\{0,-z\},\{0,z\}}\left[e^{sU+tV}W_z(u,v)(A_z(u,v)-s)^k(B_z(u,v)-t)^k\right]
=
\mathcal{R}_{\{0,-z\},\{0,z\}}\left[e^{(s-t)U}W_z(u,v)(A_z(u,v)-(s-t))^kB_z(u,v)^k\right]. \tag*{(88)} R { 0 , − z } , { 0 , z } [ e s U + t V W z ( u , v ) ( A z ( u , v ) − s ) k ( B z ( u , v ) − t ) k ] = R { 0 , − z } , { 0 , z } [ e ( s − t ) U W z ( u , v ) ( A z ( u , v ) − ( s − t ) ) k B z ( u , v ) k ] . (88) which proves the lemma.
Proof of Lemma 2.8
We use similar residue-degree bookkeeping as in the proof of Lemma 2.6 . Write the two bracketed sums in the definition of J L , r ( a ) ( σ ) J_{L,r}^{(a)}(\sigma) J L , r ( a ) ( σ )
S ( x , y ) : = ∑ j = 1 k 1 x j + ∑ ℓ = 1 k 1 y ℓ + 1 , T ( x , y ) : = ∑ j = 1 k 1 x j − 1 + ∑ ℓ = 1 k 1 y ℓ . (89) S(x,y):=\sum_{j=1}^{k}\frac{1}{x_j}+\sum_{\ell=1}^{k}\frac{1}{y_\ell+1},\qquad T(x,y):=\sum_{j=1}^{k}\frac{1}{x_j-1}+\sum_{\ell=1}^{k}\frac{1}{y_\ell}. \tag*{(89)} S ( x , y ) := j = 1 ∑ k x j 1 + ℓ = 1 ∑ k y ℓ + 1 1 , T ( x , y ) := j = 1 ∑ k x j − 1 1 + ℓ = 1 ∑ k y ℓ 1 . (89) First observe that, for ℓ ≤ r \ell\le r ℓ ≤ r y ℓ = − 1 y_\ell=-1 y ℓ = − 1 ( y ℓ + 1 ) − 1 (y_\ell+1)^{-1} ( y ℓ + 1 ) − 1 S ( x , y ) S(x,y) S ( x , y ) ( y ℓ + 1 ) − 1 (y_\ell+1)^{-1} ( y ℓ + 1 ) − 1 ℓ = 1 , … , r \ell=1,\ldots,r ℓ = 1 , … , r a < r a<r a < r J L , r ( a ) ( σ ) = 0 J_{L,r}^{(a)}(\sigma)=0 J L , r ( a ) ( σ ) = 0
Assume now that a ≥ r a\ge r a ≥ r S ( x , y ) a S(x,y)^a S ( x , y ) a D D D x j − 1 x_j^{-1} x j − 1 x r + 1 , … , x k x_{r+1},\ldots,x_k x r + 1 , … , x k H H H ( y ℓ + 1 ) − 1 (y_\ell+1)^{-1} ( y ℓ + 1 ) − 1 y 1 , … , y r y_1,\ldots,y_r y 1 , … , y r
D + H ≤ a , H ≥ r D+H\le a,\qquad H\ge r D + H ≤ a , H ≥ r for a non-zero contribution. Now consider a fixed monomial in the expansion of T ( x , y ) k T(x,y)^k T ( x , y ) k j ≤ r j\le r j ≤ r x j = 1 x_j=1 x j = 1 ( x j − 1 ) − 1 (x_j-1)^{-1} ( x j − 1 ) − 1 ( x j − 1 ) − α j (x_j-1)^{-\alpha_j} ( x j − 1 ) − α j
A : = ∑ j = 1 r α j . A:=\sum_{j=1}^{r}\alpha_j. A := j = 1 ∑ r α j . As this monomial has total degree k k k A ≤ k A\le k A ≤ k α j ≥ 1 \alpha_j\ge1 α j ≥ 1 j ≤ r j\le r j ≤ r x j = 1 x_j=1 x j = 1 x 1 , … , x r x_1,\ldots,x_r x 1 , … , x r x j = 1 x_j=1 x j = 1 ( x j − 1 ) α j − 1 (x_j-1)^{\alpha_j-1} ( x j − 1 ) α j − 1 Φ L ( σ x 1 , … , σ x k ) \Phi_L(\sigma x_1,\ldots,\sigma x_k) Φ L ( σ x 1 , … , σ x k ) x 1 − 1 , … , x r − 1 x_1-1,\ldots,x_r-1 x 1 − 1 , … , x r − 1
∑ j = 1 r ( α j − 1 ) = A − r ≤ k − r . \sum_{j=1}^{r}(\alpha_j-1)=A-r\le k-r. j = 1 ∑ r ( α j − 1 ) = A − r ≤ k − r . It remains to bound the possible degree coming from the unshifted variables x r + 1 , … , x k x_{r+1},\ldots,x_k x r + 1 , … , x k j > r j>r j > r x j − k x_j^{-k} x j − k k k k x j = 0 x_j=0 x j = 0 S ( x , y ) a S(x,y)^a S ( x , y ) a D D D x x x x r + 1 = ⋯ = x k = 0 x_{r+1}=\cdots=x_k=0 x r + 1 = ⋯ = x k = 0
( k − r ) ( k − 1 ) + D (k-r)(k-1)+D ( k − r ) ( k − 1 ) + D in the variables $x_{r+1},\ldots,x_k. For the mixed product, use the same coefficient-wise expansion as in the proof of Lemma 2.6 . For fixed ℓ ≤ r \ell\le r ℓ ≤ r
∏ j = r + 1 k ( x j − ( y ℓ + 1 ) ) = ∑ p = 0 k − r ( − 1 ) p ( y ℓ + 1 ) p e k − r − p ( x r + 1 , … , x k ) . \prod_{j=r+1}^{k}\left(x_j-(y_\ell+1)\right)=\sum_{p=0}^{k-r}(-1)^p(y_\ell+1)^p e_{k-r-p}(x_{r+1},\ldots,x_k). j = r + 1 ∏ k ( x j − ( y ℓ + 1 ) ) = p = 0 ∑ k − r ( − 1 ) p ( y ℓ + 1 ) p e k − r − p ( x r + 1 , … , x k ) . If ν ℓ \nu_\ell ν ℓ ( y ℓ + 1 ) − 1 (y_\ell+1)^{-1} ( y ℓ + 1 ) − 1 S ( x , y ) a S(x,y)^a S ( x , y ) a y ℓ = − 1 y_\ell=-1 y ℓ = − 1 x x x ν ℓ − 1 \nu_\ell-1 ν ℓ − 1 H = ∑ ℓ = 1 r ν ℓ H=\sum_{\ell=1}^{r}\nu_\ell H = ∑ ℓ = 1 r ν ℓ
r ( k − r ) − ∑ ℓ = 1 r ( ν ℓ − 1 ) = r ( k − r ) − ( H − r ) r(k-r)-\sum_{\ell=1}^{r}(\nu_\ell-1)=r(k-r)-(H-r) r ( k − r ) − ℓ = 1 ∑ r ( ν ℓ − 1 ) = r ( k − r ) − ( H − r ) in the unshifted x x x x r + 1 , … , x k x_{r+1},\ldots,x_k x r + 1 , … , x k ( k − r ) ( k − r − 1 ) (k-r)(k-r-1) ( k − r ) ( k − r − 1 )
( k − r ) ( k − r − 1 ) + r ( k − r ) − ( H − r ) = ( k − r ) ( k − 1 ) − ( H − r ) (k-r)(k-r-1)+r(k-r)-(H-r)=(k-r)(k-1)-(H-r) ( k − r ) ( k − r − 1 ) + r ( k − r ) − ( H − r ) = ( k − r ) ( k − 1 ) − ( H − r ) in x r + 1 , … , x k x_{r+1},\ldots,x_k x r + 1 , … , x k Φ L ( σ x 1 , … , σ x k ) \Phi_L(\sigma x_1,\ldots,\sigma x_k) Φ L ( σ x 1 , … , σ x k ) x r + 1 , … , x k x_{r+1},\ldots,x_k x r + 1 , … , x k
( ( k − r ) ( k − 1 ) + D ) − ( ( k − r ) ( k − 1 ) − ( H − r ) ) = D + H − r ≤ a − r . (90) \bigl((k-r)(k-1)+D\bigr)-\bigl((k-r)(k-1)-(H-r)\bigr)=D+H-r\le a-r. \tag*{(90)} ( ( k − r ) ( k − 1 ) + D ) − ( ( k − r ) ( k − 1 ) − ( H − r ) ) = D + H − r ≤ a − r . (90) We use multi-indices
β = ( β 1 , … , β r ) ∈ Z ≥ 0 r , γ = ( γ 1 , … , γ k − r ) ∈ Z ≥ 0 k − r , \beta=(\beta_1,\ldots,\beta_r)\in\mathbb{Z}_{\ge0}^{r},\qquad\gamma=(\gamma_1,\ldots,\gamma_{k-r})\in\mathbb{Z}_{\ge0}^{k-r}, β = ( β 1 , … , β r ) ∈ Z ≥ 0 r , γ = ( γ 1 , … , γ k − r ) ∈ Z ≥ 0 k − r , where β \beta β ( x 1 − 1 ) , … , ( x r − 1 ) (x_1-1),\ldots,(x_r-1) ( x 1 − 1 ) , … , ( x r − 1 ) γ \gamma γ x r + 1 , … , x k x_{r+1},\ldots,x_k x r + 1 , … , x k
∣ β ∣ = ∑ j = 1 r β j , ∣ γ ∣ = ∑ m = 1 k − r γ m , β ! = ∏ j = 1 r β j ! , γ ! = ∏ m = 1 k − r γ m ! , (91) |\beta|=\sum_{j=1}^{r}\beta_j,\qquad|\gamma|=\sum_{m=1}^{k-r}\gamma_m,\qquad\beta!=\prod_{j=1}^{r}\beta_j!,\qquad\gamma!=\prod_{m=1}^{k-r}\gamma_m!, \tag*{(91)} ∣ β ∣ = j = 1 ∑ r β j , ∣ γ ∣ = m = 1 ∑ k − r γ m , β ! = j = 1 ∏ r β j ! , γ ! = m = 1 ∏ k − r γ m ! , (91) with the usual convention that empty products equal 1. As Φ L ( σ x 1 , … , σ x k ) \Phi_L(\sigma x_1,\ldots,\sigma x_k) Φ L ( σ x 1 , … , σ x k ) x x x
Φ L ( σ x 1 , … , σ x k ) = ∑ β ∈ Z ≥ 0 r ∑ γ ∈ Z ≥ 0 k − r ϕ β , γ ( r ) ( L , σ ) ∏ j = 1 r ( x j − 1 ) β j ∏ j = r + 1 k x j γ j − r , \Phi_L(\sigma x_1,\ldots,\sigma x_k)=\sum_{\beta\in\mathbb{Z}_{\ge0}^{r}}\sum_{\gamma\in\mathbb{Z}_{\ge0}^{k-r}}\phi_{\beta,\gamma}^{(r)}(L,\sigma)\prod_{j=1}^{r}(x_j-1)^{\beta_j}\prod_{j=r+1}^{k}x_j^{\gamma_{j-r}}, Φ L ( σ x 1 , … , σ x k ) = β ∈ Z ≥ 0 r ∑ γ ∈ Z ≥ 0 k − r ∑ ϕ β , γ ( r ) ( L , σ ) j = 1 ∏ r ( x j − 1 ) β j j = r + 1 ∏ k x j γ j − r , into the definition of J L , r ( a ) ( σ ) J_{L,r}^{(a)}(\sigma) J L , r ( a ) ( σ ) ∣ β ∣ > k − r |\beta|>k-r ∣ β ∣ > k − r ∣ γ ∣ > a − r |\gamma|>a-r ∣ γ ∣ > a − r
J L , r ( a ) ( σ ) = ∑ β ∈ Z ≥ 0 r ∣ β ∣ ≤ k − r ∑ γ ∈ Z ≥ 0 k − r ∣ γ ∣ ≤ a − r ϕ β , γ ( r ) ( L , σ ) C β , γ ( r , a ) , J_{L,r}^{(a)}(\sigma)=\sum_{\substack{\beta\in\mathbb{Z}_{\ge0}^{r}\\|\beta|\le k-r}}\sum_{\substack{\gamma\in\mathbb{Z}_{\ge0}^{k-r}\\|\gamma|\le a-r}}\phi_{\beta,\gamma}^{(r)}(L,\sigma)C_{\beta,\gamma}^{(r,a)}, J L , r ( a ) ( σ ) = β ∈ Z ≥ 0 r ∣ β ∣ ≤ k − r ∑ γ ∈ Z ≥ 0 k − r ∣ γ ∣ ≤ a − r ∑ ϕ β , γ ( r ) ( L , σ ) C β , γ ( r , a ) , where the constants C β , γ ( r , a ) C_{\beta,\gamma}^{(r,a)} C β , γ ( r , a ) L L L σ \sigma σ
Proof of Theorem 2.5
Recall that I ~ k ( n ; N ) \widetilde{I}_k(n;N) I k ( n ; N ) ≤ n \le n ≤ n I k ( n ; N ) I_k(n;N) I k ( n ; N ) n n n (30) and (28) . To obtain the leading asymptotic for I ~ k ( n ; N ) \widetilde{I}_k(n;N) I k ( n ; N ) I k ( n ; N ) I_k(n;N) I k ( n ; N )
1 2 π i ∫ ∣ z ∣ = η ( z Λ U ′ Λ U ( z ) ) k 1 z n + 1 d z = 1 2 π i ∫ ∣ z ∣ = η ( − ∑ j = 1 ∞ Tr ( U ∗ j ) z j ) k 1 z n + 1 d z = ( − 1 ) k ∑ m = 1 ∞ ∑ j 1 , … , j k ≥ 1 j 1 + ⋯ + j k = m 1 2 π i ∫ ∣ z ∣ = η Tr ( U ∗ j 1 ) ⋯ Tr ( U ∗ j k ) z m − ( n + 1 ) d z = ( − 1 ) k ∑ j 1 , … , j k ≥ 1 j 1 + ⋯ + j k = n Tr ( U ∗ j 1 ) ⋯ Tr ( U ∗ j k ) , (92) \begin{aligned}
\frac{1}{2\pi i}\int_{|z|=\eta}\left(z\frac{\Lambda'_U}{\Lambda_U}(z)\right)^k\frac{1}{z^{n+1}}\,dz
=\frac{1}{2\pi i}\int_{|z|=\eta}\left(-\sum_{j=1}^{\infty}\operatorname{Tr}(U^{*j})z^j\right)^k\frac{1}{z^{n+1}}\,dz \\
&=(-1)^k\sum_{m=1}^{\infty}\sum_{\substack{j_1,\ldots,j_k\ge1\\j_1+\cdots+j_k=m}}\frac{1}{2\pi i}\int_{|z|=\eta}\operatorname{Tr}(U^{*j_1})\cdots\operatorname{Tr}(U^{*j_k})z^{m-(n+1)}\,dz \\
&=(-1)^k\sum_{\substack{j_1,\ldots,j_k\ge1\\j_1+\cdots+j_k=n}}\operatorname{Tr}(U^{*j_1})\cdots\operatorname{Tr}(U^{*j_k}), \tag*{(92)}
\end{aligned} 2 π i 1 ∫ ∣ z ∣ = η ( z Λ U Λ U ′ ( z ) ) k z n + 1 1 d z = 2 π i 1 ∫ ∣ z ∣ = η ( − j = 1 ∑ ∞ Tr ( U ∗ j ) z j ) k z n + 1 1 d z = ( − 1 ) k m = 1 ∑ ∞ j 1 , … , j k ≥ 1 j 1 + ⋯ + j k = m ∑ 2 π i 1 ∫ ∣ z ∣ = η Tr ( U ∗ j 1 ) ⋯ Tr ( U ∗ j k ) z m − ( n + 1 ) d z = ( − 1 ) k j 1 , … , j k ≥ 1 j 1 + ⋯ + j k = n ∑ Tr ( U ∗ j 1 ) ⋯ Tr ( U ∗ j k ) , (92) where ∣ η ∣ < 1 |\eta|<1 ∣ η ∣ < 1
I k ( n ; N ) : = 1 ( 2 π i ) 2 ∫ U ( N ) ∫ ∣ z 1 ∣ = η ∫ ∣ z 2 ∣ = η ( z 1 Λ U ′ Λ U ( z 1 ) ) k ( z 2 Λ U ∗ ′ Λ U ∗ ( z 2 ) ) k 1 ( z 1 z 2 ) n + 1 d z 1 d z 2 d U = ∑ j 1 , … , j k ≥ 1 j 1 + ⋯ + j k = n ∑ ℓ 1 , … , ℓ k ≥ 1 ℓ 1 + ⋯ + ℓ k = n ∫ U ( N ) Tr ( U j 1 ) ⋯ Tr ( U j k ) Tr ( U ℓ 1 ) ⋯ Tr ( U ℓ k ) ‾ d U = ∫ U ( N ) ∣ ∑ j 1 , … , j k ≥ 1 j 1 + ⋯ + j k = n Tr ( U j 1 ) ⋯ Tr ( U j k ) ∣ 2 d U . \begin{aligned}
I_k(n;N)&:=\frac{1}{(2\pi i)^2}\int_{U(N)}\int_{|z_1|=\eta}\int_{|z_2|=\eta}\left(z_1\frac{\Lambda'_U}{\Lambda_U}(z_1)\right)^k\left(z_2\frac{\Lambda'_{U^*}}{\Lambda_{U^*}}(z_2)\right)^k\frac{1}{(z_1z_2)^{n+1}}\,dz_1dz_2dU\\
&=\sum_{\substack{j_1,\ldots,j_k\ge1\\j_1+\cdots+j_k=n}}\sum_{\substack{\ell_1,\ldots,\ell_k\ge1\\\ell_1+\cdots+\ell_k=n}}\int_{U(N)}\operatorname{Tr}(U^{j_1})\cdots\operatorname{Tr}(U^{j_k})\overline{\operatorname{Tr}(U^{\ell_1})\cdots\operatorname{Tr}(U^{\ell_k})}\,dU\\
&=\int_{U(N)}\left|\sum_{\substack{j_1,\ldots,j_k\ge1\\j_1+\cdots+j_k=n}}\operatorname{Tr}(U^{j_1})\cdots\operatorname{Tr}(U^{j_k})\right|^2dU.
\end{aligned} I k ( n ; N ) := ( 2 π i ) 2 1 ∫ U ( N ) ∫ ∣ z 1 ∣ = η ∫ ∣ z 2 ∣ = η ( z 1 Λ U Λ U ′ ( z 1 ) ) k ( z 2 Λ U ∗ Λ U ∗ ′ ( z 2 ) ) k ( z 1 z 2 ) n + 1 1 d z 1 d z 2 d U = j 1 , … , j k ≥ 1 j 1 + ⋯ + j k = n ∑ ℓ 1 , … , ℓ k ≥ 1 ℓ 1 + ⋯ + ℓ k = n ∑ ∫ U ( N ) Tr ( U j 1 ) ⋯ Tr ( U j k ) Tr ( U ℓ 1 ) ⋯ Tr ( U ℓ k ) d U = ∫ U ( N ) j 1 , … , j k ≥ 1 j 1 + ⋯ + j k = n ∑ Tr ( U j 1 ) ⋯ Tr ( U j k ) 2 d U . In the above, we use the phase invariance of Haar measure as shown in obtaining (28) . Due to (27) , I k ( n ; N ) I_k(n;N) I k ( n ; N )
I k ( n ; N ) = 1 ( 2 π i ) 2 ∫ ∣ z 1 ∣ = η ∫ ∣ z 2 ∣ = η F k ( z 1 z 2 ; N ) ( z 1 z 2 ) n + 1 d z 1 d z 2 = 1 2 π i ∫ ∣ s ∣ = η 2 F k ( s ; N ) s n + 1 d s , (93) I_k(n;N)=\frac{1}{(2\pi i)^2}\int_{|z_1|=\eta}\int_{|z_2|=\eta}\frac{F_k(z_1z_2;N)}{(z_1z_2)^{n+1}}\,dz_1dz_2=\frac{1}{2\pi i}\int_{|s|=\eta^2}\frac{F_k(s;N)}{s^{n+1}}\,ds, \tag*{(93)} I k ( n ; N ) = ( 2 π i ) 2 1 ∫ ∣ z 1 ∣ = η ∫ ∣ z 2 ∣ = η ( z 1 z 2 ) n + 1 F k ( z 1 z 2 ; N ) d z 1 d z 2 = 2 π i 1 ∫ ∣ s ∣ = η 2 s n + 1 F k ( s ; N ) d s , (93) where
F k ( z 1 z 2 ; N ) = ∫ U ( N ) ( z 1 Λ U ′ Λ U ( z 1 ) ) k ( z 2 Λ U ∗ ′ Λ U ∗ ( z 2 ) ) k d U . (94) F_k(z_1z_2;N)=\int_{U(N)}\left(z_1\frac{\Lambda'_U}{\Lambda_U}(z_1)\right)^k\left(z_2\frac{\Lambda'_{U^*}}{\Lambda_{U^*}}(z_2)\right)^k\,dU. \tag*{(94)} F k ( z 1 z 2 ; N ) = ∫ U ( N ) ( z 1 Λ U Λ U ′ ( z 1 ) ) k ( z 2 Λ U ∗ Λ U ∗ ′ ( z 2 ) ) k d U . (94) We may also express I ~ k ( n ; N ) \widetilde{I}_k(n;N) I k ( n ; N ) F k ( s ; N ) F_k(s;N) F k ( s ; N )
I ~ k ( n ; N ) = ∑ j = 1 n I k ( j , N ) = − 1 2 π i ∫ ∣ ξ ∣ = η 2 F k ( ξ ; N ) ξ n + 1 ( ξ − 1 ) d ξ . (95) \widetilde{I}_k(n;N)=\sum_{j=1}^{n}I_k(j,N)=-\frac{1}{2\pi i}\int_{|\xi|=\eta^2}\frac{F_k(\xi;N)}{\xi^{n+1}(\xi-1)}\,d\xi. \tag*{(95)} I k ( n ; N ) = j = 1 ∑ n I k ( j , N ) = − 2 π i 1 ∫ ∣ ξ ∣ = η 2 ξ n + 1 ( ξ − 1 ) F k ( ξ ; N ) d ξ . (95) An expression for F k ( ξ ; N ) F_k(\xi;N) F k ( ξ ; N ) ∣ ξ ∣ < 1 |\xi|<1 ∣ ξ ∣ < 1 [8 ] ), but was rewritten as a multiple contour integral in equation (43) of [1 ] . In either case, it is derived by differentiating the Ratios Theorem (Theorem 4.1 in [7 ] ), which we restate in Theorem 3.1 . To get an expression in a form that is useful to us, we will repeat the steps of that process here.
Theorem 3.1 ([7 ] , Theorem 4.1). Let k , N ≥ 1 k,N\ge1 k , N ≥ 1
α 1 , … , α 2 k , γ 1 , … , γ k , δ 1 , … , δ k ∈ C , ℜ γ q > 0 , ℜ δ r > 0 \alpha_1,\ldots,\alpha_{2k},\gamma_1,\ldots,\gamma_k,\delta_1,\ldots,\delta_k\in\mathbb{C},\qquad\Re\gamma_q>0,\qquad\Re\delta_r>0 α 1 , … , α 2 k , γ 1 , … , γ k , δ 1 , … , δ k ∈ C , ℜ γ q > 0 , ℜ δ r > 0 for 1 ≤ q , r ≤ k 1 \le q,r \le k 1 ≤ q , r ≤ k
∫ U ( N ) ∏ j = 1 k Λ U ( e − α j ) ∏ l = k + 1 2 k Λ U ∗ ( e α l ) ∏ q = 1 k Λ U ( e − γ q ) ∏ r = 1 k Λ U ∗ ( e − δ r ) d U N = ∑ σ ∈ Ξ k , k e N ∑ p = 1 k ( α σ ( p ) − α p ) ∏ p = 1 k ∏ l = k + 1 2 k z ( α σ ( p ) − α σ ( l ) ) ∏ q = 1 k ∏ r = 1 k z ( γ q + δ r ) ∏ r = 1 k ∏ p = 1 k z ( α σ ( p ) + δ r ) ∏ q = 1 k ∏ l = k + 1 2 k z ( γ q − α σ ( l ) ) . (96) \begin{aligned}
\int_{U(N)}\frac{\prod_{j=1}^{k}\Lambda_U(e^{-\alpha_j})\prod_{l=k+1}^{2k}\Lambda_{U^*}(e^{\alpha_l})}{\prod_{q=1}^{k}\Lambda_U(e^{-\gamma_q})\prod_{r=1}^{k}\Lambda_{U^*}(e^{-\delta_r})}\,dU_N
={}&\sum_{\sigma\in\Xi_{k,k}}e^{N\sum_{p=1}^{k}(\alpha_{\sigma(p)}-\alpha_p)}
\frac{\prod_{p=1}^{k}\prod_{l=k+1}^{2k}z(\alpha_{\sigma(p)}-\alpha_{\sigma(l)})\prod_{q=1}^{k}\prod_{r=1}^{k}z(\gamma_q+\delta_r)}
{\prod_{r=1}^{k}\prod_{p=1}^{k}z(\alpha_{\sigma(p)}+\delta_r)\prod_{q=1}^{k}\prod_{l=k+1}^{2k}z(\gamma_q-\alpha_{\sigma(l)})}. \tag*{(96)}
\end{aligned} ∫ U ( N ) ∏ q = 1 k Λ U ( e − γ q ) ∏ r = 1 k Λ U ∗ ( e − δ r ) ∏ j = 1 k Λ U ( e − α j ) ∏ l = k + 1 2 k Λ U ∗ ( e α l ) d U N = σ ∈ Ξ k , k ∑ e N ∑ p = 1 k ( α σ ( p ) − α p ) ∏ r = 1 k ∏ p = 1 k z ( α σ ( p ) + δ r ) ∏ q = 1 k ∏ l = k + 1 2 k z ( γ q − α σ ( l ) ) ∏ p = 1 k ∏ l = k + 1 2 k z ( α σ ( p ) − α σ ( l ) ) ∏ q = 1 k ∏ r = 1 k z ( γ q + δ r ) . (96) where z ( x ) : = 1 1 − e − x z(x):=\frac{1}{1-e^{-x}} z ( x ) := 1 − e − x 1 Ξ k , k \Xi_{k,k} Ξ k , k σ ∈ S 2 k \sigma\in S_{2k} σ ∈ S 2 k 1 ≤ σ ( 1 ) < ⋯ < σ ( k ) ≤ 2 k 1\le\sigma(1)<\cdots<\sigma(k)\le2k 1 ≤ σ ( 1 ) < ⋯ < σ ( k ) ≤ 2 k 1 ≤ σ ( k + 1 ) < ⋯ < σ ( 2 k ) ≤ 2 k 1\le\sigma(k+1)<\cdots<\sigma(2k)\le2k 1 ≤ σ ( k + 1 ) < ⋯ < σ ( 2 k ) ≤ 2 k
Next we let a j = e α j a_j=e^{\alpha_j} a j = e α j 1 ≤ j ≤ 2 k 1\le j\le2k 1 ≤ j ≤ 2 k c q = e γ q c_q=e^{\gamma_q} c q = e γ q d r = e − δ r d_r=e^{-\delta_r} d r = e − δ r 1 ≤ q , r ≤ k 1\le q,r\le k 1 ≤ q , r ≤ k ∣ d r ∣ < 1 |d_r|<1 ∣ d r ∣ < 1 ∣ c q ∣ > 1 |c_q|>1 ∣ c q ∣ > 1
∫ U ( N ) ∏ j = 1 k Λ U ( 1 a j ) ∏ l = k + 1 2 k Λ U ∗ ( a l ) ∏ q = 1 k Λ U ( 1 c q ) ∏ r = 1 k Λ U ∗ ( d r ) d U N = ( − 1 ) k ∏ j = 1 k a j − N ∑ σ ∈ Ξ k , k ( ∏ j = 1 k a σ ( j ) N ) × ∏ r = 1 k ∏ p = 1 k ( 1 − a σ ( p ) d r ) ∏ q = 1 k ∏ l = k + 1 2 k ( a σ ( l ) − c q ) ∏ p = 1 k ∏ l = k + 1 2 k ( a σ ( p ) − a σ ( l ) ) ∏ q = 1 k ∏ r = 1 k ( 1 − c q d r ) . (97) \begin{aligned}
\int_{U(N)}\frac{\prod_{j=1}^{k}\Lambda_U\left(\frac{1}{a_j}\right)\prod_{l=k+1}^{2k}\Lambda_{U^*}(a_l)}{\prod_{q=1}^{k}\Lambda_U\left(\frac{1}{c_q}\right)\prod_{r=1}^{k}\Lambda_{U^*}(d_r)}\,dU_N
={}&(-1)^k\prod_{j=1}^{k}a_j^{-N}\sum_{\sigma\in\Xi_{k,k}}\left(\prod_{j=1}^{k}a_{\sigma(j)}^N\right)\\
&\times\frac{\prod_{r=1}^{k}\prod_{p=1}^{k}\left(1-\frac{a_{\sigma(p)}}{d_r}\right)\prod_{q=1}^{k}\prod_{l=k+1}^{2k}(a_{\sigma(l)}-c_q)}
{\prod_{p=1}^{k}\prod_{l=k+1}^{2k}(a_{\sigma(p)}-a_{\sigma(l)})\prod_{q=1}^{k}\prod_{r=1}^{k}\left(1-\frac{c_q}{d_r}\right)}. \tag*{(97)}
\end{aligned} ∫ U ( N ) ∏ q = 1 k Λ U ( c q 1 ) ∏ r = 1 k Λ U ∗ ( d r ) ∏ j = 1 k Λ U ( a j 1 ) ∏ l = k + 1 2 k Λ U ∗ ( a l ) d U N = ( − 1 ) k j = 1 ∏ k a j − N σ ∈ Ξ k , k ∑ ( j = 1 ∏ k a σ ( j ) N ) × ∏ p = 1 k ∏ l = k + 1 2 k ( a σ ( p ) − a σ ( l ) ) ∏ q = 1 k ∏ r = 1 k ( 1 − d r c q ) ∏ r = 1 k ∏ p = 1 k ( 1 − d r a σ ( p ) ) ∏ q = 1 k ∏ l = k + 1 2 k ( a σ ( l ) − c q ) . (97) In the above, we obtain an overall ( − 1 ) k = ( − 1 ) k 2 (-1)^k=(-1)^{k^2} ( − 1 ) k = ( − 1 ) k 2 z ( x ) z(x) z ( x ) (96) . We then apply Lemma 2.5.1 from [6 ] to express (97) as
∏ j = 1 k a j − N ( k ! ) 2 ( 2 π i ) 2 k ∮ ⋯ ∮ ∏ j = 1 k w j N ∏ r = 1 k ∏ p = 1 k ( 1 − w p d r ) ∏ q = 1 k ∏ l = k + 1 2 k ( w l − c q ) ∏ p = 1 k ∏ l = k + 1 2 k ( w p − w l ) ∏ q = 1 k ∏ r = 1 k ( 1 − c q d r ) ∏ 1 ≤ i < j ≤ 2 k ( w j − w i ) 2 ∏ j = 1 2 k d w j ∏ p = 1 2 k ∏ j = 1 2 k ( w p − a j ) . (98) \begin{aligned}
\frac{\prod_{j=1}^{k}a_j^{-N}}{(k!)^2(2\pi i)^{2k}}
\oint\cdots\oint
\prod_{j=1}^{k}w_j^N
\frac{\prod_{r=1}^{k}\prod_{p=1}^{k}\left(1-\frac{w_p}{d_r}\right)\prod_{q=1}^{k}\prod_{l=k+1}^{2k}(w_l-c_q)}
{\prod_{p=1}^{k}\prod_{l=k+1}^{2k}(w_p-w_l)\prod_{q=1}^{k}\prod_{r=1}^{k}\left(1-\frac{c_q}{d_r}\right)}
\frac{\prod_{1\le i<j\le2k}(w_j-w_i)^2\prod_{j=1}^{2k}dw_j}
{\prod_{p=1}^{2k}\prod_{j=1}^{2k}(w_p-a_j)}. \tag*{(98)}
\end{aligned} ( k ! ) 2 ( 2 π i ) 2 k ∏ j = 1 k a j − N ∮ ⋯ ∮ j = 1 ∏ k w j N ∏ p = 1 k ∏ l = k + 1 2 k ( w p − w l ) ∏ q = 1 k ∏ r = 1 k ( 1 − d r c q ) ∏ r = 1 k ∏ p = 1 k ( 1 − d r w p ) ∏ q = 1 k ∏ l = k + 1 2 k ( w l − c q ) ∏ p = 1 2 k ∏ j = 1 2 k ( w p − a j ) ∏ 1 ≤ i < j ≤ 2 k ( w j − w i ) 2 ∏ j = 1 2 k d w j . (98) Note that each of the w w w a j a_j a j 1 ≤ j ≤ 2 k 1\le j\le2k 1 ≤ j ≤ 2 k [1 ] , we proceed by computing a series of derivatives
d d a 1 ⋯ d d a 2 k ∫ U ( N ) ∏ j = 1 k Λ U ( 1 a j ) ∏ l = k + 1 2 k Λ U ∗ ( a l ) ∏ q = 1 k Λ U ( 1 c q ) ∏ r = 1 k Λ U ∗ ( d r ) d U N ∣ a 1 , … , a k = a a k + 1 , … , a 2 k = b c q = a , d r = b . (99) \left.
\frac{d}{da_1}\cdots\frac{d}{da_{2k}}
\int_{U(N)}
\frac{\prod_{j=1}^{k}\Lambda_U\left(\frac{1}{a_j}\right)\prod_{l=k+1}^{2k}\Lambda_{U^*}(a_l)}
{\prod_{q=1}^{k}\Lambda_U\left(\frac{1}{c_q}\right)\prod_{r=1}^{k}\Lambda_{U^*}(d_r)}
\,dU_N
\right|_{\substack{a_1,\ldots,a_k=a\\a_{k+1},\ldots,a_{2k}=b\\c_q=a,d_r=b}}. \tag*{(99)} d a 1 d ⋯ d a 2 k d ∫ U ( N ) ∏ q = 1 k Λ U ( c q 1 ) ∏ r = 1 k Λ U ∗ ( d r ) ∏ j = 1 k Λ U ( a j 1 ) ∏ l = k + 1 2 k Λ U ∗ ( a l ) d U N a 1 , … , a k = a a k + 1 , … , a 2 k = b c q = a , d r = b . (99) On the one hand, this equals
( − 1 ) k a 2 k ∫ U ( N ) Λ U ′ ( 1 / a ) k Λ U ∗ ′ ( b ) k Λ U ( 1 / a ) k Λ U ∗ ( b ) k d U N (100) \frac{(-1)^k}{a^{2k}}\int_{U(N)}
\frac{\Lambda'_U(1/a)^k\Lambda'_{U^*}(b)^k}{\Lambda_U(1/a)^k\Lambda_{U^*}(b)^k}\,dU_N \tag*{(100)} a 2 k ( − 1 ) k ∫ U ( N ) Λ U ( 1/ a ) k Λ U ∗ ( b ) k Λ U ′ ( 1/ a ) k Λ U ∗ ′ ( b ) k d U N (100) with the ( − 1 ) k / a 2 k (-1)^k/a^{2k} ( − 1 ) k / a 2 k (98) , it equals
d d a 1 ⋯ d d a 2 k ( ∏ j = 1 k a j − N ( k ! ) 2 ( 2 π i ) 2 k \left.
\frac{d}{da_1}\cdots\frac{d}{da_{2k}}
\left(
\frac{\prod_{j=1}^{k}a_j^{-N}}{(k!)^2(2\pi i)^{2k}}
\right.\right. d a 1 d ⋯ d a 2 k d ( ( k ! ) 2 ( 2 π i ) 2 k ∏ j = 1 k a j − N × ∮ ⋯ ∮ ∏ j = 1 k w j N ∏ r = 1 k ∏ p = 1 k ( 1 − w p d r ) ∏ q = 1 k ∏ l = k + 1 2 k ( w l − c q ) ∏ p = 1 k ∏ l = k + 1 2 k ( w p − w l ) ∏ q = 1 k ∏ r = 1 k ( 1 − c q d r ) ∏ 1 ≤ i < j ≤ 2 k ( w j − w i ) 2 ∏ j = 1 2 k d w j ∏ p = 1 2 k ∏ j = 1 2 k ( w p − a j ) ) ∣ a 1 , … , a k = a a k + 1 , … , a 2 k = b c q = a , d r = b . (101) \left.\left.
\times\oint\cdots\oint
\prod_{j=1}^{k}w_j^N
\frac{\prod_{r=1}^{k}\prod_{p=1}^{k}\left(1-\frac{w_p}{d_r}\right)\prod_{q=1}^{k}\prod_{l=k+1}^{2k}(w_l-c_q)}
{\prod_{p=1}^{k}\prod_{l=k+1}^{2k}(w_p-w_l)\prod_{q=1}^{k}\prod_{r=1}^{k}\left(1-\frac{c_q}{d_r}\right)}
\frac{\prod_{1\le i<j\le2k}(w_j-w_i)^2\prod_{j=1}^{2k}dw_j}
{\prod_{p=1}^{2k}\prod_{j=1}^{2k}(w_p-a_j)}
\right)
\right|_{\substack{a_1,\ldots,a_k=a\\a_{k+1},\ldots,a_{2k}=b\\c_q=a,d_r=b}}. \tag*{(101)} × ∮ ⋯ ∮ j = 1 ∏ k w j N ∏ p = 1 k ∏ l = k + 1 2 k ( w p − w l ) ∏ q = 1 k ∏ r = 1 k ( 1 − d r c q ) ∏ r = 1 k ∏ p = 1 k ( 1 − d r w p ) ∏ q = 1 k ∏ l = k + 1 2 k ( w l − c q ) ∏ p = 1 2 k ∏ j = 1 2 k ( w p − a j ) ∏ 1 ≤ i < j ≤ 2 k ( w j − w i ) 2 ∏ j = 1 2 k d w j a 1 , … , a k = a a k + 1 , … , a 2 k = b c q = a , d r = b . (101) For ε ∈ { 0 , 1 } \varepsilon\in\{0,1\} ε ∈ { 0 , 1 }
d d a ( a − ε N ∏ j = 1 2 k ( w j − a ) ) = a − ε N ∏ j = 1 2 k ( w j − a ) ( − ε N a + ∑ j = 1 2 k 1 w j − a ) . (102) \frac{d}{da}\left(\frac{a^{-\varepsilon N}}{\prod_{j=1}^{2k}(w_j-a)}\right)=\frac{a^{-\varepsilon N}}{\prod_{j=1}^{2k}(w_j-a)}\left(-\frac{\varepsilon N}{a}+\sum_{j=1}^{2k}\frac{1}{w_j-a}\right). \tag*{(102)} d a d ( ∏ j = 1 2 k ( w j − a ) a − εN ) = ∏ j = 1 2 k ( w j − a ) a − εN ( − a εN + j = 1 ∑ 2 k w j − a 1 ) . (102) For each of the first k k k a 1 , … , a k a_1,\ldots,a_k a 1 , … , a k ε = 1 \varepsilon=1 ε = 1 (102) and the remaining variables are the ε = 0 \varepsilon=0 ε = 0
( − 1 ) k a 2 k ∫ U ( N ) Λ U ′ ( 1 / a ) k Λ U ∗ ′ ( b ) k Λ U ( 1 / a ) k Λ U ∗ ( b ) k d U N = 1 ( k ! ) 2 ( 2 π i ) 2 k ∮ ⋯ ∮ ∏ j = 1 2 k w j N [ a − N ∏ j = 1 2 k ( w j − a ) ( − N a + ∑ j = 1 2 k 1 w j − a ) ] k × [ 1 ∏ j = 1 2 k ( w j − b ) ( ∑ j = 1 2 k 1 w j − b ) ] k ∏ 1 ≤ i < j ≤ 2 k ( w j − w i ) 2 × ∏ j = 1 k ( 1 − w j b ) k ∏ l = k + 1 2 k ( w l − a ) k ( ∏ j = 1 k ∏ l = k + 1 2 k ( w j − w l ) ) ( 1 − a b ) k 2 ∏ j = 1 2 k d w j . (103) \begin{aligned}
\frac{(-1)^k}{a^{2k}}\int_{U(N)}\frac{\Lambda'_U(1/a)^k\Lambda'_{U^*}(b)^k}{\Lambda_U(1/a)^k\Lambda_{U^*}(b)^k}\,dU_N
={}&\frac{1}{(k!)^2(2\pi i)^{2k}}\oint\cdots\oint\prod_{j=1}^{2k}w_j^N
\left[\frac{a^{-N}}{\prod_{j=1}^{2k}(w_j-a)}
\left(-\frac{N}{a}+\sum_{j=1}^{2k}\frac{1}{w_j-a}\right)\right]^k\\
&\times\left[\frac{1}{\prod_{j=1}^{2k}(w_j-b)}
\left(\sum_{j=1}^{2k}\frac{1}{w_j-b}\right)\right]^k
\prod_{1\le i<j\le2k}(w_j-w_i)^2\\
&\times\frac{\prod_{j=1}^k\left(1-\frac{w_j}{b}\right)^k\prod_{l=k+1}^{2k}(w_l-a)^k}
{\left(\prod_{j=1}^k\prod_{l=k+1}^{2k}(w_j-w_l)\right)\left(1-\frac{a}{b}\right)^{k^2}}
\prod_{j=1}^{2k}dw_j. \tag*{(103)}
\end{aligned} a 2 k ( − 1 ) k ∫ U ( N ) Λ U ( 1/ a ) k Λ U ∗ ( b ) k Λ U ′ ( 1/ a ) k Λ U ∗ ′ ( b ) k d U N = ( k ! ) 2 ( 2 π i ) 2 k 1 ∮ ⋯ ∮ j = 1 ∏ 2 k w j N [ ∏ j = 1 2 k ( w j − a ) a − N ( − a N + j = 1 ∑ 2 k w j − a 1 ) ] k × [ ∏ j = 1 2 k ( w j − b ) 1 ( j = 1 ∑ 2 k w j − b 1 ) ] k 1 ≤ i < j ≤ 2 k ∏ ( w j − w i ) 2 × ( ∏ j = 1 k ∏ l = k + 1 2 k ( w j − w l ) ) ( 1 − b a ) k 2 ∏ j = 1 k ( 1 − b w j ) k ∏ l = k + 1 2 k ( w l − a ) k j = 1 ∏ 2 k d w j . (103) The variables c q c_q c q d r d_r d r c q = a c_q=a c q = a d r = b d_r=b d r = b ∣ a ∣ > 1 |a|>1 ∣ a ∣ > 1 ∣ b ∣ < 1 |b|<1 ∣ b ∣ < 1 ∣ c q ∣ > 1 |c_q|>1 ∣ c q ∣ > 1 ∣ d r ∣ < 1 |d_r|<1 ∣ d r ∣ < 1 (103) , we multiply both the numerator and denominator by b k 2 b^{k^2} b k 2 ( 1 − w j / b ) (1-w_j/b) ( 1 − w j / b ) b − w j b-w_j b − w j ( 1 − a / b ) (1-a/b) ( 1 − a / b ) ( b − a ) (b-a) ( b − a )
1 a 2 k ∫ U ( N ) Λ U ′ ( 1 / a ) k Λ U ∗ ′ ( b ) k Λ U ( 1 / a ) k Λ U ∗ ( b ) k d U N = a − k N ( b − a ) k 2 ( k ! ) 2 ( 2 π i ) 2 k ∮ ⋯ ∮ ∏ j = 1 k w j N ∏ j = 1 k ( w j − b ) k ∏ l = k + 1 2 k ( w l − a ) k ( ∏ j = 1 k ∏ l = k + 1 2 k ( w j − w l ) ) ∏ j = 1 2 k ( w j − a ) k ( w j − b ) k × ( ∑ j = 1 2 k 1 w j − a − N a ) k ( ∑ j = 1 2 k 1 w j − b ) k ∏ 1 ≤ i < j ≤ 2 k ( w j − w i ) 2 ∏ j = 1 2 k d w j . (104) \begin{aligned}
\frac{1}{a^{2k}}\int_{U(N)}\frac{\Lambda'_U(1/a)^k\Lambda'_{U^*}(b)^k}{\Lambda_U(1/a)^k\Lambda_{U^*}(b)^k}\,dU_N
={}&\frac{a^{-kN}}{(b-a)^{k^2}(k!)^2(2\pi i)^{2k}}
\oint\cdots\oint
\frac{\prod_{j=1}^k w_j^N\prod_{j=1}^k(w_j-b)^k\prod_{l=k+1}^{2k}(w_l-a)^k}
{\left(\prod_{j=1}^k\prod_{l=k+1}^{2k}(w_j-w_l)\right)\prod_{j=1}^{2k}(w_j-a)^k(w_j-b)^k}\\
&\times\left(\sum_{j=1}^{2k}\frac{1}{w_j-a}-\frac{N}{a}\right)^k
\left(\sum_{j=1}^{2k}\frac{1}{w_j-b}\right)^k
\prod_{1\le i<j\le2k}(w_j-w_i)^2\prod_{j=1}^{2k}dw_j. \tag*{(104)}
\end{aligned} a 2 k 1 ∫ U ( N ) Λ U ( 1/ a ) k Λ U ∗ ( b ) k Λ U ′ ( 1/ a ) k Λ U ∗ ′ ( b ) k d U N = ( b − a ) k 2 ( k ! ) 2 ( 2 π i ) 2 k a − k N ∮ ⋯ ∮ ( ∏ j = 1 k ∏ l = k + 1 2 k ( w j − w l ) ) ∏ j = 1 2 k ( w j − a ) k ( w j − b ) k ∏ j = 1 k w j N ∏ j = 1 k ( w j − b ) k ∏ l = k + 1 2 k ( w l − a ) k × ( j = 1 ∑ 2 k w j − a 1 − a N ) k ( j = 1 ∑ 2 k w j − b 1 ) k 1 ≤ i < j ≤ 2 k ∏ ( w j − w i ) 2 j = 1 ∏ 2 k d w j . (104) Note that the w w w max ( ∣ a ∣ , ∣ b ∣ ) = ∣ a ∣ \max(|a|,|b|)=|a| max ( ∣ a ∣ , ∣ b ∣ ) = ∣ a ∣ u j = w j − a u_j=w_j-a u j = w j − a j = 1 , … , k j=1,\ldots,k j = 1 , … , k v j = w k + j − b v_j=w_{k+j}-b v j = w k + j − b j = 1 , … , k j=1,\ldots,k j = 1 , … , k u u u − a -a − a v v v − b -b − b
1 a 2 k ∮ U ( N ) Λ U ′ ( 1 / a ) k Λ U ∗ ′ ( b ) k Λ U ( 1 / a ) k Λ U ∗ ( b ) k d U N = a − k N ( b − a ) k 2 ( k ! ) 2 ( 2 π i ) 2 k ∮ ⋯ ∮ ∏ j = 1 k ( u j + a ) N ( ∏ j = 1 k ∏ l = 1 k ( u j + a − v l − b ) ) ∏ j = 1 k u j k v j k ( ∑ j = 1 k 1 u j + ∑ j = 1 k 1 v j + b − a − N a ) k ( ∑ j = 1 k 1 v j + ∑ j = 1 k 1 u j + a − b ) k Δ ( u ) 2 Δ ( v ) 2 ∏ j = 1 k ∏ l = 1 k ( v l + b − u j − a ) 2 ∏ j = 1 k d u j d v j , (105) \begin{aligned}
\frac{1}{a^{2k}}\oint_{U(N)}\frac{\Lambda'_U(1/a)^k\Lambda'_{U^*}(b)^k}{\Lambda_U(1/a)^k\Lambda_{U^*}(b)^k}\,dU_N \\
&=\frac{a^{-kN}}{(b-a)^{k^2}(k!)^2(2\pi i)^{2k}}\oint\cdots\oint
\frac{\prod_{j=1}^{k}(u_j+a)^N}{\left(\prod_{j=1}^{k}\prod_{l=1}^{k}(u_j+a-v_l-b)\right)\prod_{j=1}^{k}u_j^k v_j^k}
\left(\sum_{j=1}^{k}\frac{1}{u_j}+\sum_{j=1}^{k}\frac{1}{v_j+b-a}-\frac{N}{a}\right)^k
\left(\sum_{j=1}^{k}\frac{1}{v_j}+\sum_{j=1}^{k}\frac{1}{u_j+a-b}\right)^k
\Delta(u)^2\Delta(v)^2\prod_{j=1}^{k}\prod_{l=1}^{k}(v_l+b-u_j-a)^2\prod_{j=1}^{k}du_jdv_j, \tag*{(105)}
\end{aligned} a 2 k 1 ∮ U ( N ) Λ U ( 1/ a ) k Λ U ∗ ( b ) k Λ U ′ ( 1/ a ) k Λ U ∗ ′ ( b ) k d U N = ( b − a ) k 2 ( k ! ) 2 ( 2 π i ) 2 k a − k N ∮ ⋯ ∮ ( ∏ j = 1 k ∏ l = 1 k ( u j + a − v l − b ) ) ∏ j = 1 k u j k v j k ∏ j = 1 k ( u j + a ) N ( j = 1 ∑ k u j 1 + j = 1 ∑ k v j + b − a 1 − a N ) k ( j = 1 ∑ k v j 1 + j = 1 ∑ k u j + a − b 1 ) k Δ ( u ) 2 Δ ( v ) 2 j = 1 ∏ k l = 1 ∏ k ( v l + b − u j − a ) 2 j = 1 ∏ k d u j d v j , (105) where Δ ( u ) = ∏ i < j ( u j − u i ) \Delta(u)=\prod_{i<j}(u_j-u_i) Δ ( u ) = ∏ i < j ( u j − u i ) ( v ℓ + b − u j − a ) 2 (v_\ell+b-u_j-a)^2 ( v ℓ + b − u j − a ) 2 ( u j + a − v l − b ) (u_j+a-v_l-b) ( u j + a − v l − b ) ( − 1 ) k (-1)^k ( − 1 ) k u j + a − v l − b = 0 u_j+a-v_l-b=0 u j + a − v l − b = 0 u j u_j u j b − a b-a b − a v j v_j v j a − b a-b a − b
1 a 2 k ∮ U ( N ) Λ U ′ ( 1 / a ) k Λ U ∗ ′ ( b ) k Λ U ( 1 / a ) k Λ U ∗ ( b ) k d U N = ( − 1 ) k a − k N ( b − a ) k 2 ( k ! ) 2 ( 2 π i ) 2 k ∮ ⋯ ∮ ∏ j = 1 k ( u j + a ) N Δ ( u ) 2 Δ ( v ) 2 ∏ j = 1 k ∏ l = 1 k ( v l − u j + b − a ) ( ∑ j = 1 k 1 u j + ∑ j = 1 k 1 v j + b − a − N a ) k ( ∑ j = 1 k 1 v j + ∑ j = 1 k 1 u j + a − b ) k 1 ∏ j = 1 k u j k v j k ∏ j = 1 k d u j d v j . (106) \begin{aligned}
\frac{1}{a^{2k}}\oint_{U(N)}\frac{\Lambda'_U(1/a)^k\Lambda'_{U^*}(b)^k}{\Lambda_U(1/a)^k\Lambda_{U^*}(b)^k}\,dU_N \\
&=\frac{(-1)^k a^{-kN}}{(b-a)^{k^2}(k!)^2(2\pi i)^{2k}}\oint\cdots\oint\prod_{j=1}^{k}(u_j+a)^N\Delta(u)^2\Delta(v)^2
\prod_{j=1}^{k}\prod_{l=1}^{k}(v_l-u_j+b-a)
\left(\sum_{j=1}^{k}\frac{1}{u_j}+\sum_{j=1}^{k}\frac{1}{v_j+b-a}-\frac{N}{a}\right)^k
\left(\sum_{j=1}^{k}\frac{1}{v_j}+\sum_{j=1}^{k}\frac{1}{u_j+a-b}\right)^k
\frac{1}{\prod_{j=1}^{k}u_j^kv_j^k}\prod_{j=1}^{k}du_jdv_j. \tag*{(106)}
\end{aligned} a 2 k 1 ∮ U ( N ) Λ U ( 1/ a ) k Λ U ∗ ( b ) k Λ U ′ ( 1/ a ) k Λ U ∗ ′ ( b ) k d U N = ( b − a ) k 2 ( k ! ) 2 ( 2 π i ) 2 k ( − 1 ) k a − k N ∮ ⋯ ∮ j = 1 ∏ k ( u j + a ) N Δ ( u ) 2 Δ ( v ) 2 j = 1 ∏ k l = 1 ∏ k ( v l − u j + b − a ) ( j = 1 ∑ k u j 1 + j = 1 ∑ k v j + b − a 1 − a N ) k ( j = 1 ∑ k v j 1 + j = 1 ∑ k u j + a − b 1 ) k ∏ j = 1 k u j k v j k 1 j = 1 ∏ k d u j d v j . (106) We now scale every u u u v v v a a a ξ = b a \xi=\frac{b}{a} ξ = a b a − 2 k a^{-2k} a − 2 k
∮ U ( N ) Λ U ′ ( 1 / a ) k Λ U ∗ ′ ( b ) k Λ U ( 1 / a ) k Λ U ∗ ( b ) k d U N = ( − 1 ) k ( ξ − 1 ) k 2 ( k ! ) 2 ( 2 π i ) 2 k ∮ ⋯ ∮ ∏ j = 1 k ( u j + 1 ) N Δ ( u ) 2 Δ ( v ) 2 ∏ j = 1 k ∏ l = 1 k ( v l − u j + ξ − 1 ) ( ∑ j = 1 k 1 u j + ∑ j = 1 k 1 v j + ξ − 1 − N ) k ( ∑ j = 1 k 1 v j + ∑ j = 1 k 1 u j + 1 − ξ ) k 1 ∏ j = 1 k ( u j v j ) k ∏ j = 1 k d u j d v j , (107) \begin{aligned}
\oint_{U(N)}\frac{\Lambda'_U(1/a)^k\Lambda'_{U^*}(b)^k}{\Lambda_U(1/a)^k\Lambda_{U^*}(b)^k}\,dU_N \\
&=\frac{(-1)^k}{(\xi-1)^{k^2}(k!)^2(2\pi i)^{2k}}\oint\cdots\oint\prod_{j=1}^{k}(u_j+1)^N\Delta(u)^2\Delta(v)^2
\prod_{j=1}^{k}\prod_{l=1}^{k}(v_l-u_j+\xi-1)
\left(\sum_{j=1}^{k}\frac{1}{u_j}+\sum_{j=1}^{k}\frac{1}{v_j+\xi-1}-N\right)^k
\left(\sum_{j=1}^{k}\frac{1}{v_j}+\sum_{j=1}^{k}\frac{1}{u_j+1-\xi}\right)^k
\frac{1}{\prod_{j=1}^{k}(u_jv_j)^k}\prod_{j=1}^{k}du_jdv_j, \tag*{(107)}
\end{aligned} ∮ U ( N ) Λ U ( 1/ a ) k Λ U ∗ ( b ) k Λ U ′ ( 1/ a ) k Λ U ∗ ′ ( b ) k d U N = ( ξ − 1 ) k 2 ( k ! ) 2 ( 2 π i ) 2 k ( − 1 ) k ∮ ⋯ ∮ j = 1 ∏ k ( u j + 1 ) N Δ ( u ) 2 Δ ( v ) 2 j = 1 ∏ k l = 1 ∏ k ( v l − u j + ξ − 1 ) ( j = 1 ∑ k u j 1 + j = 1 ∑ k v j + ξ − 1 1 − N ) k ( j = 1 ∑ k v j 1 + j = 1 ∑ k u j + 1 − ξ 1 ) k ∏ j = 1 k ( u j v j ) k 1 j = 1 ∏ k d u j d v j , (107) where ∣ ξ ∣ = ∣ b / a ∣ < 1 |\xi|=|b/a|<1 ∣ ξ ∣ = ∣ b / a ∣ < 1 ∣ a ∣ > 1 |a|>1 ∣ a ∣ > 1 ∣ b ∣ < 1 |b|<1 ∣ b ∣ < 1 u j u_j u j ξ − 1 \xi-1 ξ − 1 v ℓ v_\ell v ℓ 1 − ξ 1-\xi 1 − ξ u u u − 1 -1 − 1 v v v − ξ -\xi − ξ 0 < τ < 1 / 2 0<\tau<1/2 0 < τ < 1/2 ∣ u j ∣ = ∣ v ℓ ∣ = μ |u_j|=|v_\ell|=\mu ∣ u j ∣ = ∣ v ℓ ∣ = μ μ > ( 1 + τ ) ∣ ξ − 1 ∣ \mu>(1+\tau)|\xi-1| μ > ( 1 + τ ) ∣ ξ − 1∣ (94) , for ∣ ξ ∣ < 1 |\xi|<1 ∣ ξ ∣ < 1
F k ( ξ ; N ) = ( − 1 ) k ξ k 1 ( ξ − 1 ) k 2 ( k ! ) 2 ( 2 π i ) 2 k ∫ ∣ u j ∣ = μ ∫ ∣ v ℓ ∣ = μ ∏ j = 1 k ( u j + 1 ) N Δ ( u ) 2 Δ ( v ) 2 ∏ j , ℓ = 1 k ( v ℓ − u j + ξ − 1 ) ( ∑ j = 1 k 1 u j + ∑ j = 1 k 1 v j + ξ − 1 − N ) k ( ∑ j = 1 k 1 v j + ∑ j = 1 k 1 u j − ( ξ − 1 ) ) k 1 ∏ j = 1 k u j k v j k ∏ j = 1 k d u j d v j . (108) F_k(\xi;N)=(-1)^k\xi^k\frac{1}{(\xi-1)^{k^2}(k!)^2(2\pi i)^{2k}}\int_{|u_j|=\mu}\int_{|v_\ell|=\mu}\prod_{j=1}^k (u_j+1)^N\Delta(u)^2\Delta(v)^2
\prod_{j,\ell=1}^k(v_\ell-u_j+\xi-1)
\left(\sum_{j=1}^k\frac{1}{u_j}+\sum_{j=1}^k\frac{1}{v_j+\xi-1}-N\right)^k
\left(\sum_{j=1}^k\frac{1}{v_j}+\sum_{j=1}^k\frac{1}{u_j-(\xi-1)}\right)^k
\frac{1}{\prod_{j=1}^k u_j^k v_j^k}\prod_{j=1}^kdu_jdv_j. \tag*{(108)} F k ( ξ ; N ) = ( − 1 ) k ξ k ( ξ − 1 ) k 2 ( k ! ) 2 ( 2 π i ) 2 k 1 ∫ ∣ u j ∣ = μ ∫ ∣ v ℓ ∣ = μ j = 1 ∏ k ( u j + 1 ) N Δ ( u ) 2 Δ ( v ) 2 j , ℓ = 1 ∏ k ( v ℓ − u j + ξ − 1 ) ( j = 1 ∑ k u j 1 + j = 1 ∑ k v j + ξ − 1 1 − N ) k ( j = 1 ∑ k v j 1 + j = 1 ∑ k u j − ( ξ − 1 ) 1 ) k ∏ j = 1 k u j k v j k 1 j = 1 ∏ k d u j d v j . (108) Since ∏ j , ℓ = 1 k ( v ℓ − u j + ξ − 1 ) = ( − 1 ) k 2 ∏ j , ℓ = 1 k ( u j − v ℓ − ( ξ − 1 ) ) \prod_{j,\ell=1}^k(v_\ell-u_j+\xi-1)=(-1)^{k^2}\prod_{j,\ell=1}^k(u_j-v_\ell-(\xi-1)) ∏ j , ℓ = 1 k ( v ℓ − u j + ξ − 1 ) = ( − 1 ) k 2 ∏ j , ℓ = 1 k ( u j − v ℓ − ( ξ − 1 )) ( − 1 ) k (-1)^k ( − 1 ) k F k ( ξ ; N ) F_k(\xi;N) F k ( ξ ; N ) L = N L=N L = N σ = ξ − 1 \sigma=\xi-1 σ = ξ − 1 Φ N ( u 1 , … , u k ) = ∏ j = 1 k ( 1 + u j ) N \Phi_N(u_1,\ldots,u_k)=\prod_{j=1}^k(1+u_j)^N Φ N ( u 1 , … , u k ) = ∏ j = 1 k ( 1 + u j ) N
F k ( ξ ; N ) = ξ k ( 2 π i ) 2 k ( k ! ) 2 ∑ r = 0 k ( k r ) 2 ∑ a = 0 k ( − 1 ) k − a ( k a ) N k − a ( ξ − 1 ) k + a J N , r ( a ) ( ξ − 1 ) , (109) F_k(\xi;N)=\frac{\xi^k}{(2\pi i)^{2k}(k!)^2}\sum_{r=0}^k\binom{k}{r}^2\sum_{a=0}^k(-1)^{k-a}\binom{k}{a}\frac{N^{k-a}}{(\xi-1)^{k+a}}J_{N,r}^{(a)}(\xi-1), \tag*{(109)} F k ( ξ ; N ) = ( 2 π i ) 2 k ( k ! ) 2 ξ k r = 0 ∑ k ( r k ) 2 a = 0 ∑ k ( − 1 ) k − a ( a k ) ( ξ − 1 ) k + a N k − a J N , r ( a ) ( ξ − 1 ) , (109) with
J N , r ( a ) ( ξ − 1 ) = ∫ c j ( r ) ∫ c ^ ℓ ( r ) ∏ j = 1 k ( 1 + ( ξ − 1 ) x j ) N Δ ( x ) 2 Δ ( y ) 2 ∏ j , ℓ = 1 k ( x j − y ℓ − 1 ) ( ∑ j = 1 k 1 x j + ∑ ℓ = 1 k 1 y ℓ + 1 ) ( ∑ j = 1 k 1 x j − 1 + ∑ ℓ = 1 k 1 y ℓ ) a ∏ j = 1 k x j − k d x j ∏ ℓ = 1 k y ℓ − k d y ℓ . J_{N,r}^{(a)}(\xi-1)=\int_{c_j(r)}\int_{\hat c_\ell(r)}\prod_{j=1}^k(1+(\xi-1)x_j)^N\Delta(x)^2\Delta(y)^2\prod_{j,\ell=1}^k(x_j-y_\ell-1)
\left(\sum_{j=1}^k\frac{1}{x_j}+\sum_{\ell=1}^k\frac{1}{y_\ell+1}\right)
\left(\sum_{j=1}^k\frac{1}{x_j-1}+\sum_{\ell=1}^k\frac{1}{y_\ell}\right)^a
\prod_{j=1}^k x_j^{-k}dx_j\prod_{\ell=1}^k y_\ell^{-k}dy_\ell. J N , r ( a ) ( ξ − 1 ) = ∫ c j ( r ) ∫ c ^ ℓ ( r ) j = 1 ∏ k ( 1 + ( ξ − 1 ) x j ) N Δ ( x ) 2 Δ ( y ) 2 j , ℓ = 1 ∏ k ( x j − y ℓ − 1 ) ( j = 1 ∑ k x j 1 + ℓ = 1 ∑ k y ℓ + 1 1 ) ( j = 1 ∑ k x j − 1 1 + ℓ = 1 ∑ k y ℓ 1 ) a j = 1 ∏ k x j − k d x j ℓ = 1 ∏ k y ℓ − k d y ℓ . Here, c j ( r ) c_j(r) c j ( r ) τ \tau τ 1 1 1 j = 1 , … , r j=1,\ldots,r j = 1 , … , r 0 0 0 c ^ ℓ ( r ) \hat c_\ell(r) c ^ ℓ ( r ) τ \tau τ − 1 -1 − 1 ℓ = 1 , … , r \ell=1,\ldots,r ℓ = 1 , … , r x 1 , … , x r x_1,\ldots,x_r x 1 , … , x r 1 1 1 y 1 , … , y r y_1,\ldots,y_r y 1 , … , y r − 1 -1 − 1 0 0 0 ξ \xi ξ r r r j ≤ r j\le r j ≤ r x j = 1 x_j=1 x j = 1
1 + ( ξ − 1 ) x j = 1 + ( ξ − 1 ) ( 1 + ( x j − 1 ) ) = ξ + ( ξ − 1 ) ( x j − 1 ) , 1+(\xi-1)x_j=1+(\xi-1)(1+(x_j-1))=\xi+(\xi-1)(x_j-1), 1 + ( ξ − 1 ) x j = 1 + ( ξ − 1 ) ( 1 + ( x j − 1 )) = ξ + ( ξ − 1 ) ( x j − 1 ) , and therefore
( 1 + ( ξ − 1 ) x j ) N = ∑ β j = 0 N ( N β j ) ξ N − β j ( ξ − 1 ) β j ( x j − 1 ) β j . (1+(\xi-1)x_j)^N=\sum_{\beta_j=0}^N\binom{N}{\beta_j}\xi^{N-\beta_j}(\xi-1)^{\beta_j}(x_j-1)^{\beta_j}. ( 1 + ( ξ − 1 ) x j ) N = β j = 0 ∑ N ( β j N ) ξ N − β j ( ξ − 1 ) β j ( x j − 1 ) β j . We use the convention ( N β ) \binom{N}{\beta} ( β N ) N < β N<\beta N < β j = r + 1 , … , k j=r+1,\ldots,k j = r + 1 , … , k x j = 0 x_j=0 x j = 0
( 1 + ( ξ − 1 ) x j ) N = ∑ γ j − r = 0 N ( N γ j − r ) ( ξ − 1 ) γ j − r x j γ j − r . (1+(\xi-1)x_j)^N=\sum_{\gamma_{j-r}=0}^N\binom{N}{\gamma_{j-r}}(\xi-1)^{\gamma_{j-r}}x_j^{\gamma_{j-r}}. ( 1 + ( ξ − 1 ) x j ) N = γ j − r = 0 ∑ N ( γ j − r N ) ( ξ − 1 ) γ j − r x j γ j − r . Combining the two expansions above, we can write
∏ j = 1 k ( 1 + ( ξ − 1 ) x j ) N = ∑ β ∈ Z ≥ 0 r ∑ γ ∈ Z ≥ 0 k − r ϕ β , γ ( r ) ( N , ξ − 1 ) ∏ j = 1 r ( x j − 1 ) β j ∏ j = r + 1 k x j γ j − r . \prod_{j=1}^k(1+(\xi-1)x_j)^N=\sum_{\boldsymbol{\beta}\in\mathbb{Z}_{\geq0}^r}\sum_{\boldsymbol{\gamma}\in\mathbb{Z}_{\geq0}^{k-r}}\phi_{\boldsymbol{\beta},\boldsymbol{\gamma}}^{(r)}(N,\xi-1)\prod_{j=1}^r(x_j-1)^{\beta_j}\prod_{j=r+1}^k x_j^{\gamma_{j-r}}. j = 1 ∏ k ( 1 + ( ξ − 1 ) x j ) N = β ∈ Z ≥ 0 r ∑ γ ∈ Z ≥ 0 k − r ∑ ϕ β , γ ( r ) ( N , ξ − 1 ) j = 1 ∏ r ( x j − 1 ) β j j = r + 1 ∏ k x j γ j − r . where
ϕ β , γ ( r ) ( N , ξ − 1 ) = ( ∏ j = 1 r ( N β j ) ) ( ∏ m = 1 k − r ( N γ m ) ) ξ N r − ∣ β ∣ ( ξ − 1 ) ∣ β ∣ + ∣ γ ∣ . (110) \phi_{\beta,\gamma}^{(r)}(N,\xi-1)=\left(\prod_{j=1}^{r}\binom{N}{\beta_j}\right)\left(\prod_{m=1}^{k-r}\binom{N}{\gamma_m}\right)\xi^{Nr-|\beta|}(\xi-1)^{|\beta|+|\gamma|}. \tag*{(110)} ϕ β , γ ( r ) ( N , ξ − 1 ) = ( j = 1 ∏ r ( β j N ) ) ( m = 1 ∏ k − r ( γ m N ) ) ξ N r − ∣ β ∣ ( ξ − 1 ) ∣ β ∣ + ∣ γ ∣ . (110) Now refer to the definitions from (91) ; every non-zero term has β j ≤ N \beta_j\leq N β j ≤ N 1 ≤ j ≤ r 1\leq j\leq r 1 ≤ j ≤ r ∣ β ∣ ≤ N r |\beta|\leq Nr ∣ β ∣ ≤ N r ξ N r − ∣ β ∣ \xi^{Nr-|\beta|} ξ N r − ∣ β ∣ Lemma 2.8 , we have
J N , r ( a ) ( ξ − 1 ) = 0 ( a < r ) , (111) J_{N,r}^{(a)}(\xi-1)=0 \qquad(a<r), \tag*{(111)} J N , r ( a ) ( ξ − 1 ) = 0 ( a < r ) , (111) and, for a ≥ r a\geq r a ≥ r
J N , r ( a ) ( ξ − 1 ) = ∑ β ∈ Z ≥ 0 r ∣ β ∣ ≤ k − r ∑ γ ∈ Z ≥ 0 k − r ∣ γ ∣ ≤ a − r C β , γ ( r , a ) ( ∏ j = 1 r ( N β j ) ) ( ∏ m = 1 k − r ( N γ m ) ) ξ N r − ∣ β ∣ ( ξ − 1 ) ∣ β ∣ + ∣ γ ∣ , (112) J_{N,r}^{(a)}(\xi-1)=\sum_{\substack{\beta\in\mathbb{Z}_{\geq0}^{r}\\|\beta|\leq k-r}}\sum_{\substack{\gamma\in\mathbb{Z}_{\geq0}^{k-r}\\|\gamma|\leq a-r}}C_{\beta,\gamma}^{(r,a)}\left(\prod_{j=1}^{r}\binom{N}{\beta_j}\right)\left(\prod_{m=1}^{k-r}\binom{N}{\gamma_m}\right)\xi^{Nr-|\beta|}(\xi-1)^{|\beta|+|\gamma|}, \tag*{(112)} J N , r ( a ) ( ξ − 1 ) = β ∈ Z ≥ 0 r ∣ β ∣ ≤ k − r ∑ γ ∈ Z ≥ 0 k − r ∣ γ ∣ ≤ a − r ∑ C β , γ ( r , a ) ( j = 1 ∏ r ( β j N ) ) ( m = 1 ∏ k − r ( γ m N ) ) ξ N r − ∣ β ∣ ( ξ − 1 ) ∣ β ∣ + ∣ γ ∣ , (112) where
C β , γ ( r , a ) : = ∫ C j ( r ) ∫ C ^ ℓ ( r ) ∏ j = 1 r ( x j − 1 ) β j ∏ j = r + 1 k x j γ j − r Δ ( x ) 2 Δ ( y ) 2 ∏ j , ℓ = 1 k ( x j − y ℓ − 1 ) × ( ∑ j = 1 k 1 x j + ∑ ℓ = 1 k 1 y ℓ + 1 ) a ( ∑ j = 1 k 1 x j − 1 + ∑ ℓ = 1 k 1 y ℓ ) ∏ j = 1 k x j − k d x j ∏ ℓ = 1 k y ℓ − k d y ℓ . (113) \begin{aligned}
C_{\beta,\gamma}^{(r,a)}:= {}&\int_{C_j(r)}\int_{\widehat C_\ell(r)}
\prod_{j=1}^{r}(x_j-1)^{\beta_j}
\prod_{j=r+1}^{k}x_j^{\gamma_{j-r}}
\Delta(x)^2\Delta(y)^2
\prod_{j,\ell=1}^{k}(x_j-y_\ell-1)\\
&\times\left(\sum_{j=1}^{k}\frac{1}{x_j}+\sum_{\ell=1}^{k}\frac{1}{y_\ell+1}\right)^a
\left(\sum_{j=1}^{k}\frac{1}{x_j-1}+\sum_{\ell=1}^{k}\frac{1}{y_\ell}\right)
\prod_{j=1}^{k}x_j^{-k}\,dx_j\prod_{\ell=1}^{k}y_\ell^{-k}\,dy_\ell. \tag*{(113)}
\end{aligned} C β , γ ( r , a ) := ∫ C j ( r ) ∫ C ℓ ( r ) j = 1 ∏ r ( x j − 1 ) β j j = r + 1 ∏ k x j γ j − r Δ ( x ) 2 Δ ( y ) 2 j , ℓ = 1 ∏ k ( x j − y ℓ − 1 ) × ( j = 1 ∑ k x j 1 + ℓ = 1 ∑ k y ℓ + 1 1 ) a ( j = 1 ∑ k x j − 1 1 + ℓ = 1 ∑ k y ℓ 1 ) j = 1 ∏ k x j − k d x j ℓ = 1 ∏ k y ℓ − k d y ℓ . (113) We substitute (109) and (112) into
I ~ k ( n ; N ) = − 1 2 π i ∫ ∣ ξ ∣ = η F k ( ξ ; N ) ξ n + 1 ( ξ − 1 ) d ξ . (114) \widetilde I_k(n;N)=-\frac{1}{2\pi i}\int_{|\xi|=\eta}\frac{F_k(\xi;N)}{\xi^{n+1}(\xi-1)}\,d\xi. \tag*{(114)} I k ( n ; N ) = − 2 π i 1 ∫ ∣ ξ ∣ = η ξ n + 1 ( ξ − 1 ) F k ( ξ ; N ) d ξ . (114) For each admissible pair ( β , γ ) (\beta,\gamma) ( β , γ )
1 2 π i ∫ ∣ ξ ∣ = η ξ N r − n + k − 1 − ∣ β ∣ ( ξ − 1 ) − d d ξ , d : = a + k + 1 − ∣ β ∣ − ∣ γ ∣ . (115) \frac{1}{2\pi i}\int_{|\xi|=\eta}\xi^{Nr-n+k-1-|\beta|}(\xi-1)^{-d}\,d\xi,\qquad d:=a+k+1-|\beta|-|\gamma|. \tag*{(115)} 2 π i 1 ∫ ∣ ξ ∣ = η ξ N r − n + k − 1 − ∣ β ∣ ( ξ − 1 ) − d d ξ , d := a + k + 1 − ∣ β ∣ − ∣ γ ∣. (115) The degree bounds imply d > 0 d>0 d > 0 r ≥ 1 r\geq1 r ≥ 1 d ≥ 2 r + 1 d\geq2r+1 d ≥ 2 r + 1 r = 0 r=0 r = 0 d ≥ k + 1 d\geq k+1 d ≥ k + 1 ∣ ξ ∣ < 1 |\xi|<1 ∣ ξ ∣ < 1
( ξ − 1 ) − d = ( − 1 ) d ( 1 − ξ ) − d = ( − 1 ) d ∑ q ≥ 0 ( q + d − 1 d − 1 ) ξ q , (116) (\xi-1)^{-d}=(-1)^d(1-\xi)^{-d}=(-1)^d\sum_{q\geq0}\binom{q+d-1}{d-1}\xi^q, \tag*{(116)} ( ξ − 1 ) − d = ( − 1 ) d ( 1 − ξ ) − d = ( − 1 ) d q ≥ 0 ∑ ( d − 1 q + d − 1 ) ξ q , (116) and Cauchy’s integral formula gives the exact identity
1 2 π i ∫ ∣ ξ ∣ = η ξ N r − n + k − 1 − ∣ β ∣ ( ξ − 1 ) d d ξ = ( − 1 ) d ( n − N r − k + ∣ β ∣ + d − 1 d − 1 ) 1 { n − N r − k + ∣ β ∣ ≥ 0 } . (117) \frac{1}{2\pi i}\int_{|\xi|=\eta}\frac{\xi^{Nr-n+k-1-|\beta|}}{(\xi-1)^d}\,d\xi=(-1)^d\binom{n-Nr-k+|\beta|+d-1}{d-1}\mathbb{1}_{\{n-Nr-k+|\beta|\geq0\}}. \tag*{(117)} 2 π i 1 ∫ ∣ ξ ∣ = η ( ξ − 1 ) d ξ N r − n + k − 1 − ∣ β ∣ d ξ = ( − 1 ) d ( d − 1 n − N r − k + ∣ β ∣ + d − 1 ) 1 { n − N r − k + ∣ β ∣ ≥ 0 } . (117) It remains to identify the leading coefficient for these terms. For fixed β \beta β γ \gamma γ
∏ j = 1 r ( N β j ) ∏ m = 1 k − r ( N γ m ) = N ∣ β ∣ + ∣ γ ∣ β ! γ ! + O ( N ∣ β ∣ + ∣ γ ∣ − 1 ) . (118) \prod_{j=1}^{r}\binom{N}{\beta_j}\prod_{m=1}^{k-r}\binom{N}{\gamma_m}=\frac{N^{|\beta|+|\gamma|}}{\beta!\,\gamma!}+O\left(N^{|\beta|+|\gamma|-1}\right). \tag*{(118)} j = 1 ∏ r ( β j N ) m = 1 ∏ k − r ( γ m N ) = β ! γ ! N ∣ β ∣ + ∣ γ ∣ + O ( N ∣ β ∣ + ∣ γ ∣ − 1 ) . (118) We can isolate the role of the parameter r r r (112) into the outer integral (95) and applying Cauchy’s theorem, the contribution of the fixed pair ( β , γ ) (\beta,\gamma) ( β , γ )
n − N r − k + ∣ β ∣ ≥ 0. (119) n-Nr-k+|\beta|\geq0. \tag*{(119)} n − N r − k + ∣ β ∣ ≥ 0. (119) In particular, if r > α r>\alpha r > α n − N r = ( α − r ) N + O ( 1 ) n-Nr=(\alpha-r)N+O(1) n − N r = ( α − r ) N + O ( 1 ) − ∞ -\infty − ∞ ∣ β ∣ |\beta| ∣ β ∣ k k k (119) fails for all sufficiently large N N N r > α r>\alpha r > α r < α r<\alpha r < α (119) is satisfied for every admissible β \beta β N N N r = α r=\alpha r = α α \alpha α (117) is O ( 1 ) O(1) O ( 1 ) n − N α = O ( 1 ) n-N\alpha=O(1) n − N α = O ( 1 ) N k − a N^{k-a} N k − a (109) and using (118) , we have at most N k − a + ∣ β ∣ + ∣ γ ∣ = O ( N 2 k − 2 ) N^{k-a+|\beta|+|\gamma|}=O(N^{2k-2}) N k − a + ∣ β ∣ + ∣ γ ∣ = O ( N 2 k − 2 ) ∣ β ∣ ≤ k − r |\beta| \le k-r ∣ β ∣ ≤ k − r ∣ γ ∣ ≤ a − r |\gamma| \le a-r ∣ γ ∣ ≤ a − r Lemma 2.8 . Therefore, we will see shortly that the boundary term is lower order and for the leading term, we proceed with only r r r r < α r<\alpha r < α
Since r < α r<\alpha r < α
( n − N r − k + ∣ β ∣ + d − 1 d − 1 ) = N d − 1 ( α − r ) d − 1 ( d − 1 ) ! + O ( N d − 2 ) . (120) \binom{n-Nr-k+|\beta|+d-1}{d-1}=\frac{N^{d-1}(\alpha-r)^{d-1}}{(d-1)!}+O(N^{d-2}). \tag*{(120)} ( d − 1 n − N r − k + ∣ β ∣ + d − 1 ) = ( d − 1 )! N d − 1 ( α − r ) d − 1 + O ( N d − 2 ) . (120) As ∣ β ∣ + ∣ γ ∣ + d − 1 = a + k |\beta|+|\gamma|+d-1=a+k ∣ β ∣ + ∣ γ ∣ + d − 1 = a + k N a + k N^{a+k} N a + k N k − a N^{k-a} N k − a (109) , each surviving ( r , a ) (r,a) ( r , a ) N 2 k N^{2k} N 2 k β \beta β γ \gamma γ q : = d − 1 = a + k − ∣ β ∣ − ∣ γ ∣ q:=d-1=a+k-|\beta|-|\gamma| q := d − 1 = a + k − ∣ β ∣ − ∣ γ ∣ ∏ j = 1 r ( x j − 1 ) β j ∏ j = r + 1 k x j γ j − r \prod_{j=1}^{r}(x_j-1)^{\beta_j}\prod_{j=r+1}^{k}x_j^{\gamma_{j-r}} ∏ j = 1 r ( x j − 1 ) β j ∏ j = r + 1 k x j γ j − r
( − 1 ) q ( α − r ) q q ! β ! γ ! . (121) \frac{(-1)^q(\alpha-r)^q}{q!\,\beta!\,\gamma!}. \tag*{(121)} q ! β ! γ ! ( − 1 ) q ( α − r ) q . (121) On the other hand,
∑ j = 1 k x j − α = ∑ j = 1 r ( x j − 1 ) + ∑ j = r + 1 k x j − ( α − r ) . (122) \sum_{j=1}^{k}x_j-\alpha=\sum_{j=1}^{r}(x_j-1)+\sum_{j=r+1}^{k}x_j-(\alpha-r). \tag*{(122)} j = 1 ∑ k x j − α = j = 1 ∑ r ( x j − 1 ) + j = r + 1 ∑ k x j − ( α − r ) . (122) Therefore, by the multinomial theorem,
∑ q , β , γ ≥ 0 q + ∣ β ∣ + ∣ γ ∣ = a + k ( − 1 ) q ( α − r ) q q ! β ! γ ! ∏ j = 1 r ( x j − 1 ) β j ∏ j = r + 1 k x j γ j − r = ( ∑ j = 1 k x j − α ) a + k ( a + k ) ! . (123) \sum_{\substack{q,\beta,\gamma\ge0\\q+|\beta|+|\gamma|=a+k}}\frac{(-1)^q(\alpha-r)^q}{q!\,\beta!\,\gamma!}\prod_{j=1}^{r}(x_j-1)^{\beta_j}\prod_{j=r+1}^{k}x_j^{\gamma_{j-r}}=\frac{\left(\sum_{j=1}^{k}x_j-\alpha\right)^{a+k}}{(a+k)!}. \tag*{(123)} q , β , γ ≥ 0 q + ∣ β ∣ + ∣ γ ∣ = a + k ∑ q ! β ! γ ! ( − 1 ) q ( α − r ) q j = 1 ∏ r ( x j − 1 ) β j j = r + 1 ∏ k x j γ j − r = ( a + k )! ( ∑ j = 1 k x j − α ) a + k . (123) Due to the multinomial sum collapse above and using the leading-order forms of both (118) and (120) , the 2 k 2k 2 k J N , r a ( ξ − 1 ) J_{N,r}^{a}(\xi-1) J N , r a ( ξ − 1 ) (112) is simplified greatly. As the ξ \xi ξ
I ~ k ( n ; N ) = N 2 k ( 2 π i ) 2 k k ! 2 ∑ r = 0 k 1 { r < α } ( k r ) 2 ∑ a = r k ( − 1 ) k − a ( k a ) 1 ( a + k ) ! × ∫ c j ( r ) ∫ c ^ ℓ ( r ) ( ∑ j = 1 k x j − α ) a + k ( ∑ j = 1 k 1 x j + ∑ ℓ = 1 k 1 y ℓ + 1 ) a ( ∑ j = 1 k 1 x j − 1 + ∑ ℓ = 1 k 1 y ℓ ) k × Δ ( x ) 2 Δ ( y ) 2 ∏ j , ℓ = 1 k ( x j − y ℓ − 1 ) ∏ j = 1 k x j − k d x j ∏ ℓ = 1 k y ℓ − k d y ℓ + O ( N 2 k − 1 ) . (124) \begin{aligned}
\widetilde{I}_k(n;N)={}&\frac{N^{2k}}{(2\pi i)^{2k}k!^2}\sum_{r=0}^{k}\mathbf{1}_{\{r<\alpha\}}\binom{k}{r}^{2}\sum_{a=r}^{k}(-1)^{k-a}\binom{k}{a}\frac{1}{(a+k)!}\\
&\times\int_{c_j(r)}\int_{\widehat{c}_{\ell}(r)}\left(\sum_{j=1}^{k}x_j-\alpha\right)^{a+k}\left(\sum_{j=1}^{k}\frac{1}{x_j}+\sum_{\ell=1}^{k}\frac{1}{y_\ell+1}\right)^a\left(\sum_{j=1}^{k}\frac{1}{x_j-1}+\sum_{\ell=1}^{k}\frac{1}{y_\ell}\right)^k\\
&\times\Delta(x)^2\Delta(y)^2\prod_{j,\ell=1}^{k}(x_j-y_\ell-1)\prod_{j=1}^{k}x_j^{-k}\,dx_j\prod_{\ell=1}^{k}y_\ell^{-k}\,dy_\ell+O(N^{2k-1}). \tag*{(124)}
\end{aligned} I k ( n ; N ) = ( 2 π i ) 2 k k ! 2 N 2 k r = 0 ∑ k 1 { r < α } ( r k ) 2 a = r ∑ k ( − 1 ) k − a ( a k ) ( a + k )! 1 × ∫ c j ( r ) ∫ c ℓ ( r ) ( j = 1 ∑ k x j − α ) a + k ( j = 1 ∑ k x j 1 + ℓ = 1 ∑ k y ℓ + 1 1 ) a ( j = 1 ∑ k x j − 1 1 + ℓ = 1 ∑ k y ℓ 1 ) k × Δ ( x ) 2 Δ ( y ) 2 j , ℓ = 1 ∏ k ( x j − y ℓ − 1 ) j = 1 ∏ k x j − k d x j ℓ = 1 ∏ k y ℓ − k d y ℓ + O ( N 2 k − 1 ) . (124) Finally, dividing by N 2 k N^{2k} N 2 k M k ( α ) M_k(\alpha) M k ( α ) (35) yields the stated result in the theorem.
Proof of Theorem 2.2
Let k ≥ 1 k\ge1 k ≥ 1 α \alpha α α \alpha α Conjecture 1 throughout with the error decay exponent θ k , c \theta_{k,c} θ k , c
θ k , c = { 1 + c , k = 1 , 2 , k − 1 , k ≥ 3 , (125) \theta_{k,c}=
\begin{cases}
1+c, & k=1,2,\\
k-1, & k\ge3,
\end{cases} \tag*{(125)} θ k , c = { 1 + c , k − 1 , k = 1 , 2 , k ≥ 3 , (125) for some fixed 0 < c ≤ 1 0<c\le1 0 < c ≤ 1 k = 1 , 2 k=1,2 k = 1 , 2 0 < ξ < c 0<\xi<c 0 < ξ < c Λ ∗ k = Λ ∗ ⋯ ∗ Λ \Lambda^{*k}=\Lambda*\cdots*\Lambda Λ ∗ k = Λ ∗ ⋯ ∗ Λ
Λ ∗ k ( n ) = ∑ n 1 ⋯ n k = n Λ ( n 1 ) ⋯ Λ ( n k ) . \Lambda^{*k}(n)=\sum_{n_1\cdots n_k=n}\Lambda(n_1)\cdots\Lambda(n_k). Λ ∗ k ( n ) = n 1 ⋯ n k = n ∑ Λ ( n 1 ) ⋯ Λ ( n k ) . For any ordered factorisation n = n 1 ⋯ n k n=n_1\cdots n_k n = n 1 ⋯ n k 0 ≤ Λ ( n j ) ≤ log n j ≤ log n 0\leq\Lambda(n_j)\leq\log n_j\leq\log n 0 ≤ Λ ( n j ) ≤ log n j ≤ log n ( log n ) k (\log n)^k ( log n ) k k k k ( n 1 , … , n k ) (n_1,\ldots,n_k) ( n 1 , … , n k ) n n n d k ( n ) d_k(n) d k ( n ) k k k
∣ Λ ∗ k ( n ) ∣ ≤ ( log n ) k d k ( n ) ( n ≥ 2 ) . (126) \lvert\Lambda^{*k}(n)\rvert\leq(\log n)^k d_k(n)\qquad(n\geq2). \tag*{(126)} ∣ Λ ∗ k ( n )∣ ≤ ( log n ) k d k ( n ) ( n ≥ 2 ) . (126) To obtain a truncated Perron formula for
∑ n ≤ ⌊ T α ⌋ Λ ∗ k ( n ) n 1 / 2 + i t , \sum_{n\leq\lfloor T^\alpha\rfloor}\frac{\Lambda^{*k}(n)}{n^{1/2+it}}, n ≤ ⌊ T α ⌋ ∑ n 1/2 + i t Λ ∗ k ( n ) , we apply the Lemma stated below.
Lemma 3.2 (Titchmarsh [32 ] , Lemma 3.12). Let
f ( s ) = ∑ n = 1 ∞ a n n s ( ℜ s > 1 ) , f(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}\qquad(\Re s>1), f ( s ) = n = 1 ∑ ∞ n s a n ( ℜ s > 1 ) , with a n = O { ψ ( n ) } a_n=O\{\psi(n)\} a n = O { ψ ( n )} ψ ( n ) \psi(n) ψ ( n )
∑ n = 1 ∞ ∣ a n ∣ n σ = O { 1 ( σ − 1 ) μ } \sum_{n=1}^{\infty}\frac{\lvert a_n\rvert}{n^\sigma}=O\left\{\frac{1}{(\sigma-1)^\mu}\right\} n = 1 ∑ ∞ n σ ∣ a n ∣ = O { ( σ − 1 ) μ 1 } as σ → 1 + \sigma\to1^+ σ → 1 + s = σ + i t s=\sigma+it s = σ + i t c > 0 c>0 c > 0 σ + c > 1 \sigma+c>1 σ + c > 1 x x x L L L x x x
∑ n < x a n n s = 1 2 π i ∫ c − i T c + i T f ( s + w ) x w w d w + O { x c T ( σ + c − 1 ) μ } + O { ψ ( 2 x ) x 1 − σ log x T } + O { ψ ( L ) x 1 − σ T ∣ x − L ∣ } . \sum_{n<x}\frac{a_n}{n^s}=\frac{1}{2\pi i}\int_{c-iT}^{c+iT}f(s+w)\frac{x^w}{w}\,dw+O\left\{\frac{x^c}{T(\sigma+c-1)^\mu}\right\}+O\left\{\frac{\psi(2x)x^{1-\sigma}\log x}{T}\right\}+O\left\{\frac{\psi(L)x^{1-\sigma}}{T\lvert x-L\rvert}\right\}. n < x ∑ n s a n = 2 π i 1 ∫ c − i T c + i T f ( s + w ) w x w d w + O { T ( σ + c − 1 ) μ x c } + O { T ψ ( 2 x ) x 1 − σ log x } + O { T ∣ x − L ∣ ψ ( L ) x 1 − σ } . The abscissa of absolute convergence is σ a = 1 \sigma_a=1 σ a = 1
− ζ ′ ( σ ) ζ ( σ ) ≍ ( σ − 1 ) − 1 ( 1 < σ ≤ 2 ) , (127) -\frac{\zeta'(\sigma)}{\zeta(\sigma)}\asymp(\sigma-1)^{-1}\qquad(1<\sigma\leq2), \tag*{(127)} − ζ ( σ ) ζ ′ ( σ ) ≍ ( σ − 1 ) − 1 ( 1 < σ ≤ 2 ) , (127) meaning that the two quantities are bounded above and below by positive constant multiples of one another uniformly in this range, we have
∑ n ≥ 1 ∣ Λ ∗ k ( n ) ∣ n σ = ( ∑ n ≥ 1 Λ ( n ) n σ ) k = ( − ζ ′ ( σ ) ζ ( σ ) ) k ≪ k ( σ − 1 ) − k . (128) \sum_{n\geq1}\frac{\lvert\Lambda^{*k}(n)\rvert}{n^\sigma}=\left(\sum_{n\geq1}\frac{\Lambda(n)}{n^\sigma}\right)^k=\left(-\frac{\zeta'(\sigma)}{\zeta(\sigma)}\right)^k\ll_k(\sigma-1)^{-k}. \tag*{(128)} n ≥ 1 ∑ n σ ∣ Λ ∗ k ( n )∣ = ( n ≥ 1 ∑ n σ Λ ( n ) ) k = ( − ζ ( σ ) ζ ′ ( σ ) ) k ≪ k ( σ − 1 ) − k . (128) Thus, the series bound condition of Lemma 3.2 holds with μ = k \mu=k μ = k
B ( x ) : = max m ≤ x , m ∈ Z + ( log m ) k d k ( m ) , x ≥ 1 , x ∈ R , (129) B(x):=\max_{\substack{m\leq x,\ m\in\mathbb{Z}_+}}(\log m)^k d_k(m),\qquad x\geq1,\ x\in\mathbb{R}, \tag*{(129)} B ( x ) := m ≤ x , m ∈ Z + max ( log m ) k d k ( m ) , x ≥ 1 , x ∈ R , (129) for which the standard bound d k ( m ) ≪ k , ε 1 m ε 1 d_k(m)\ll_{k,\varepsilon_1}m^{\varepsilon_1} d k ( m ) ≪ k , ε 1 m ε 1 ε 1 > 0 \varepsilon_1>0 ε 1 > 0 [21 ] yields B ( x ) ≪ k , ε 1 ( log x ) k x ε 1 B(x)\ll_{k,\varepsilon_1}(\log x)^k x^{\varepsilon_1} B ( x ) ≪ k , ε 1 ( log x ) k x ε 1 T T T o ( 1 ) o(1) o ( 1 ) α \alpha α α < 2 \alpha<2 α < 2 α ( 1 / 2 + ε 1 ) < 1 \alpha(1/2+\varepsilon_1)<1 α ( 1/2 + ε 1 ) < 1 ε > 0 \varepsilon>0 ε > 0
γ : = max ( 1 , α / 2 ) + ε , T ′ : = T γ . (130) \gamma:=\max(1,\alpha/2)+\varepsilon,\qquad T':=T^\gamma. \tag*{(130)} γ := max ( 1 , α /2 ) + ε , T ′ := T γ . (130) Fixing σ = 1 2 \sigma=\frac12 σ = 2 1 κ : = 1 2 + 1 log T ′ \kappa:=\frac12+\frac{1}{\log T'} κ := 2 1 + l o g T ′ 1
∑ n < N Λ ∗ k ( n ) n 1 / 2 + i t = 1 2 π i ∫ κ − i T ′ κ + i T ′ ( − ζ ′ ( 1 2 + i t + w ) ζ ( 1 2 + i t + w ) ) k N w w d w + E t r u n c ( t ) . (131) \sum_{n<N}\frac{\Lambda^{*k}(n)}{n^{1/2+it}}=\frac{1}{2\pi i}\int_{\kappa-iT'}^{\kappa+iT'}\left(-\frac{\zeta'(\frac12+it+w)}{\zeta(\frac12+it+w)}\right)^k\frac{N^w}{w}\,dw+E_{\mathrm{trunc}}(t). \tag*{(131)} n < N ∑ n 1/2 + i t Λ ∗ k ( n ) = 2 π i 1 ∫ κ − i T ′ κ + i T ′ ( − ζ ( 2 1 + i t + w ) ζ ′ ( 2 1 + i t + w ) ) k w N w d w + E trunc ( t ) . (131) with
E t r u n c ( t ) ≪ k N κ T ′ ( σ + κ − 1 ) k + B ( 2 N ) N 1 − σ log N T ′ + B ( ⌊ N ⌋ ) N 1 − σ T ′ ∣ N − ⌊ N ⌋ ∣ . (132) E_{\mathrm{trunc}}(t) \ll_k \frac{N^\kappa}{T'(\sigma+\kappa-1)^k}+\frac{B(2N)N^{1-\sigma}\log N}{T'}+\frac{B(\lfloor N\rfloor)N^{1-\sigma}}{T'|N-\lfloor N\rfloor|}. \tag*{(132)} E trunc ( t ) ≪ k T ′ ( σ + κ − 1 ) k N κ + T ′ B ( 2 N ) N 1 − σ log N + T ′ ∣ N − ⌊ N ⌋ ∣ B (⌊ N ⌋) N 1 − σ . (132) Here we set N = ⌊ T α ⌋ + 1 2 N=\lfloor T^\alpha\rfloor+\frac{1}{2} N = ⌊ T α ⌋ + 2 1 L = ⌊ N ⌋ L=\lfloor N\rfloor L = ⌊ N ⌋ (132) . Writing N = ( T ′ ) β N=(T')^\beta N = ( T ′ ) β
β : = log N log T ′ = α γ + O ( 1 log T ) = α max ( 1 , α / 2 ) + ε + O ( 1 log T ) < 2 \beta:=\frac{\log N}{\log T'}=\frac{\alpha}{\gamma}+O\left(\frac{1}{\log T}\right)=\frac{\alpha}{\max(1,\alpha/2)+\varepsilon}+O\left(\frac{1}{\log T}\right)<2 β := log T ′ log N = γ α + O ( log T 1 ) = max ( 1 , α /2 ) + ε α + O ( log T 1 ) < 2 for all sufficiently large T T T σ + κ − 1 = 1 / log T ′ \sigma+\kappa-1=1/\log T' σ + κ − 1 = 1/ log T ′
E t r u n c ( t ) ≪ k ( log T ′ ) k ( T ′ ) β / 2 − 1 + B ( 2 N ) ( log N ) ( T ′ ) β / 2 − 1 + B ( N ) ( T ′ ) β / 2 − 1 , (133) E_{\mathrm{trunc}}(t)\ll_k(\log T')^k(T')^{\beta/2-1}+B(2N)(\log N)(T')^{\beta/2-1}+B(N)(T')^{\beta/2-1}, \tag*{(133)} E trunc ( t ) ≪ k ( log T ′ ) k ( T ′ ) β /2 − 1 + B ( 2 N ) ( log N ) ( T ′ ) β /2 − 1 + B ( N ) ( T ′ ) β /2 − 1 , (133) where the factor N 1 / log T ′ = e β N^{1/\log T'}=e^\beta N 1/ l o g T ′ = e β B ( N ) ≤ B ( 2 N ) ≪ k , ε 1 ( log N ) k N ε 1 B(N)\le B(2N)\ll_{k,\varepsilon_1}(\log N)^kN^{\varepsilon_1} B ( N ) ≤ B ( 2 N ) ≪ k , ε 1 ( log N ) k N ε 1
E t r u n c ( t ) ≪ k , ε 1 , α , ε ( log T ′ ) k ( T ′ ) β / 2 − 1 + ( log T ′ ) k + 1 ( T ′ ) β / 2 − 1 + β ε 1 E_{\mathrm{trunc}}(t)\ll_{k,\varepsilon_1,\alpha,\varepsilon}(\log T')^k(T')^{\beta/2-1}+(\log T')^{k+1}(T')^{\beta/2-1+\beta\varepsilon_1} E trunc ( t ) ≪ k , ε 1 , α , ε ( log T ′ ) k ( T ′ ) β /2 − 1 + ( log T ′ ) k + 1 ( T ′ ) β /2 − 1 + β ε 1 for any fixed ε 1 > 0 \varepsilon_1>0 ε 1 > 0 lim sup T → ∞ β < 2 \limsup_{T\to\infty}\beta<2 lim sup T → ∞ β < 2 ε 1 = ε 1 ( α , ε ) > 0 \varepsilon_1=\varepsilon_1(\alpha,\varepsilon)>0 ε 1 = ε 1 ( α , ε ) > 0
β ( 1 2 + ε 1 ) < 1 \beta\left(\frac{1}{2}+\varepsilon_1\right)<1 β ( 2 1 + ε 1 ) < 1 for all sufficiently large T T T ε ′ = ε ε , ε 1 ′ ( α ) > 0 \varepsilon'=\varepsilon'_{\varepsilon,\varepsilon_1}(\alpha)>0 ε ′ = ε ε , ε 1 ′ ( α ) > 0 T T T
β 2 − 1 ≤ − ε ′ and β 2 − 1 + β ε 1 ≤ − ε ′ . \frac{\beta}{2}-1\le-\varepsilon'\qquad\text{and}\qquad\frac{\beta}{2}-1+\beta\varepsilon_1\le-\varepsilon'. 2 β − 1 ≤ − ε ′ and 2 β − 1 + β ε 1 ≤ − ε ′ . Therefore
∑ n < N Λ ∗ k ( n ) n 1 / 2 + i t = 1 2 π i ∫ κ − i T ′ κ + i T ′ ( − ζ ′ ζ ( 1 2 + i t + w ) ) k N w w d w + E t r u n c ( t ) , \sum_{n<N}\frac{\Lambda^{*k}(n)}{n^{1/2+it}}=\frac{1}{2\pi i}\int_{\kappa-iT'}^{\kappa+iT'}\left(-\frac{\zeta'}{\zeta}\left(\frac{1}{2}+it+w\right)\right)^k\frac{N^w}{w}\,dw+E_{\mathrm{trunc}}(t), n < N ∑ n 1/2 + i t Λ ∗ k ( n ) = 2 π i 1 ∫ κ − i T ′ κ + i T ′ ( − ζ ζ ′ ( 2 1 + i t + w ) ) k w N w d w + E trunc ( t ) , where
E t r u n c ( t ) ≪ k , α , ε , ε 1 ( log T ′ ) k + 1 ( T ′ ) − ε ′ , 0 ≤ t ≤ T . (134) E_{\mathrm{trunc}}(t)\ll_{k,\alpha,\varepsilon,\varepsilon_1}(\log T')^{k+1}(T')^{-\varepsilon'},\qquad0\le t\le T. \tag*{(134)} E trunc ( t ) ≪ k , α , ε , ε 1 ( log T ′ ) k + 1 ( T ′ ) − ε ′ , 0 ≤ t ≤ T . (134) Next, we choose a contour on the conjectural scale. Fix 0 < η < 3 2 0<\eta<\frac{3}{2} 0 < η < 2 3
σ 0 : = η log T . \sigma_0:=\frac{\eta}{\log T}. σ 0 := log T η . Then, for all sufficiently large T T T 0 < σ 0 < 1 2 < κ 0<\sigma_0<\frac{1}{2}<\kappa 0 < σ 0 < 2 1 < κ ℜ w = σ 0 \Re w=\sigma_0 ℜ w = σ 0 ± T ′ \pm T' ± T ′ w = 1 2 − i t w=\frac{1}{2}-it w = 2 1 − i t − ζ ′ / ζ ( s ) -\zeta'/\zeta(s) − ζ ′ / ζ ( s ) s = 1 s=1 s = 1 ℜ w = 1 2 \Re w=\frac{1}{2} ℜ w = 2 1 T ′ > ∣ t ∣ T'>|t| T ′ > ∣ t ∣ ρ \rho ρ ζ ( s ) \zeta(s) ζ ( s ) w = ρ − ( 1 2 + i t ) w=\rho-\left(\frac{1}{2}+it\right) w = ρ − ( 2 1 + i t ) ℜ w = 0 \Re w=0 ℜ w = 0 ℜ w = σ 0 > 0 \Re w=\sigma_0>0 ℜ w = σ 0 > 0 w = 0 w=0 w = 0 ζ ( s ) \zeta(s) ζ ( s ) ℜ w = σ 0 \Re w=\sigma_0 ℜ w = σ 0
1 2 π i ∫ κ − i T ′ κ + i T ′ ( − ζ ′ ζ ( 1 2 + i t + z ) ) k N z z d z = R e s w = 1 [ ( − ζ ′ ζ ( w ) ) k N w − 1 2 − i t w − 1 2 − i t ] + 1 2 π i ∫ σ 0 − i T ′ σ 0 + i T ′ ( − ζ ′ ζ ( 1 2 + i t + z ) ) k N z z d z + E h o r ( t ) . (135) \begin{aligned}
\frac{1}{2\pi i}\int_{\kappa-iT'}^{\kappa+iT'}\left(-\frac{\zeta'}{\zeta}\left(\frac{1}{2}+it+z\right)\right)^k\frac{N^z}{z}\,dz=\mathop{\mathrm{Res}}_{w=1}\left[\left(-\frac{\zeta'}{\zeta}(w)\right)^k\frac{N^{w-\frac{1}{2}-it}}{w-\frac{1}{2}-it}\right] \\
&+\frac{1}{2\pi i}\int_{\sigma_0-iT'}^{\sigma_0+iT'}\left(-\frac{\zeta'}{\zeta}\left(\frac{1}{2}+it+z\right)\right)^k\frac{N^z}{z}\,dz+E_{\mathrm{hor}}(t). \tag*{(135)}
\end{aligned} 2 π i 1 ∫ κ − i T ′ κ + i T ′ ( − ζ ζ ′ ( 2 1 + i t + z ) ) k z N z d z = Res w = 1 [ ( − ζ ζ ′ ( w ) ) k w − 2 1 − i t N w − 2 1 − i t ] + 2 π i 1 ∫ σ 0 − i T ′ σ 0 + i T ′ ( − ζ ζ ′ ( 2 1 + i t + z ) ) k z N z d z + E hor ( t ) . (135) To estimate the horizontal contribution, E h o r ( t ) E_{\mathrm{hor}}(t) E hor ( t ) z = u ± i T ′ z=u\pm iT' z = u ± i T ′ σ 0 ≤ u ≤ κ \sigma_0\le u\le\kappa σ 0 ≤ u ≤ κ
1 2 + i t + z = 1 2 + u + i ( t ± T ′ ) . \frac{1}{2}+it+z=\frac{1}{2}+u+i(t\pm T'). 2 1 + i t + z = 2 1 + u + i ( t ± T ′ ) . Since T ′ = T γ T'=T^\gamma T ′ = T γ γ > 1 \gamma>1 γ > 1 T / T ′ → 0 T/T'\to0 T / T ′ → 0 ∣ t ± T ′ ∣ ≍ T ′ |t\pm T'|\asymp T' ∣ t ± T ′ ∣ ≍ T ′ 0 ≤ t ≤ T 0\le t\le T 0 ≤ t ≤ T
1 2 + σ 0 ≤ ℜ ( 1 2 + i t + z ) ≤ 1 + 1 log T ′ . \frac{1}{2}+\sigma_0\le\Re\left(\frac{1}{2}+it+z\right)\le1+\frac{1}{\log T'}. 2 1 + σ 0 ≤ ℜ ( 2 1 + i t + z ) ≤ 1 + log T ′ 1 . By Titchmarsh’s formula (14.15.2) in [32 ] , under RH, and using u ≥ η / log T u\ge\eta/\log T u ≥ η / log T
ζ ′ ζ ( 1 2 + i t + z ) ≪ η , α , ε ( log T ′ ) 2 (136) \frac{\zeta'}{\zeta}\left(\frac{1}{2}+it+z\right)\ll_{\eta,\alpha,\varepsilon}(\log T')^2 \tag*{(136)} ζ ζ ′ ( 2 1 + i t + z ) ≪ η , α , ε ( log T ′ ) 2 (136) on the horizontal edges. As ∣ z ∣ ≍ T ′ |z|\asymp T' ∣ z ∣ ≍ T ′ N ℜ z ≤ N κ N^{\Re z}\le N^\kappa N ℜ z ≤ N κ
E h o r ( t ) ≪ k , η , α , ε N κ ( log T ′ ) 2 k T ′ (137) E_{\mathrm{hor}}(t)\ll_{k,\eta,\alpha,\varepsilon}\frac{N^\kappa(\log T')^{2k}}{T'} \tag*{(137)} E hor ( t ) ≪ k , η , α , ε T ′ N κ ( log T ′ ) 2 k (137) uniformly for 0 ≤ t ≤ T 0\le t\le T 0 ≤ t ≤ T N κ = N 1 2 + 1 log T ′ = N 1 / 2 N 1 / log T ′ N^\kappa=N^{\frac{1}{2}+\frac{1}{\log T'}}=N^{1/2}N^{1/\log T'} N κ = N 2 1 + l o g T ′ 1 = N 1/2 N 1/ l o g T ′
N 1 / log T ′ = exp ( log N log T ′ ) = e β , N^{1/\log T'}=\exp\left(\frac{\log N}{\log T'}\right)=e^\beta, N 1/ l o g T ′ = exp ( log T ′ log N ) = e β , we have N κ = e β N 1 / 2 N^\kappa=e^\beta N^{1/2} N κ = e β N 1/2 e β = O ( 1 ) e^\beta=O(1) e β = O ( 1 ) N κ = T α / 2 + o ( 1 ) N^\kappa=T^{\alpha/2+o(1)} N κ = T α /2 + o ( 1 ) T ′ − 1 = T − γ T'^{-1}=T^{-\gamma} T ′ − 1 = T − γ
sup 0 ≤ t ≤ T ∣ E h o r ( t ) ∣ ≪ k , η , α , ε ( log T ′ ) 2 k T α / 2 − γ + o ( 1 ) . (138) \sup_{0\le t\le T}|E_{\mathrm{hor}}(t)|\ll_{k,\eta,\alpha,\varepsilon}(\log T')^{2k}T^{\alpha/2-\gamma+o(1)}. \tag*{(138)} 0 ≤ t ≤ T sup ∣ E hor ( t ) ∣ ≪ k , η , α , ε ( log T ′ ) 2 k T α /2 − γ + o ( 1 ) . (138) Since
γ − α 2 = max ( 1 − α / 2 , 0 ) + ε ≥ ε , (139) \gamma-\frac{\alpha}{2}=\max(1-\alpha/2,0)+\varepsilon\ge\varepsilon, \tag*{(139)} γ − 2 α = max ( 1 − α /2 , 0 ) + ε ≥ ε , (139) the exponent of T T T T T T k , η , α , ε k,\eta,\alpha,\varepsilon k , η , α , ε ε 1 \varepsilon_1 ε 1 E ( t ) = E h o r ( t ) + E t r u n c ( t ) E(t)=E_{\mathrm{hor}}(t)+E_{\mathrm{trunc}}(t) E ( t ) = E hor ( t ) + E trunc ( t )
∣ E ( t ) ∣ ≪ M , M : = ( log T ′ ) 2 k T α / 2 − γ + o ( 1 ) + ( log T ′ ) k + 1 T ′ − ε ′ . (140) |E(t)|\ll\mathcal{M},\qquad\mathcal{M}:=(\log T')^{2k}T^{\alpha/2-\gamma+o(1)}+(\log T')^{k+1}T'^{-\varepsilon'}. \tag*{(140)} ∣ E ( t ) ∣ ≪ M , M := ( log T ′ ) 2 k T α /2 − γ + o ( 1 ) + ( log T ′ ) k + 1 T ′ − ε ′ . (140) Since log T ′ ≍ α , ε log T \log T'\asymp_{\alpha,\varepsilon}\log T log T ′ ≍ α , ε log T
M ≪ ( log T ) 2 k T − δ , \mathcal{M}\ll(\log T)^{2k}T^{-\delta}, M ≪ ( log T ) 2 k T − δ , where
δ : = 1 2 min ( γ − α 2 , γ ε ′ ) > 0. (141) \delta:=\frac{1}{2}\min\left(\gamma-\frac{\alpha}{2},\gamma\varepsilon'\right)>0. \tag*{(141)} δ := 2 1 min ( γ − 2 α , γ ε ′ ) > 0. (141) Here the factor γ − α / 2 \gamma-\alpha/2 γ − α /2 E h o r ( t ) E_{\mathrm{hor}}(t) E hor ( t ) (138) , while γ ε ′ \gamma\varepsilon' γ ε ′ (134) upon writing ( T ′ ) − ε ′ = T − γ ε ′ (T')^{-\varepsilon'}=T^{-\gamma\varepsilon'} ( T ′ ) − ε ′ = T − γ ε ′ 1 2 \frac{1}{2} 2 1 o ( 1 ) o(1) o ( 1 ) T T T α > 0 \alpha>0 α > 0 δ \delta δ
V ( t ) : = 1 2 π i ∫ σ 0 − i T ′ σ 0 + i T ′ ( − ζ ′ ζ ( 1 2 + i t + z ) ) k N z z d z , I : = 1 T ∫ 0 T ∣ V ( t ) ∣ 2 d t . (142) V(t):=\frac{1}{2\pi i}\int_{\sigma_0-iT'}^{\sigma_0+iT'}\left(-\frac{\zeta'}{\zeta}\left(\frac{1}{2}+it+z\right)\right)^k\frac{N^z}{z}\,dz,\qquad\mathcal{I}:=\frac{1}{T}\int_0^T|V(t)|^2\,dt. \tag*{(142)} V ( t ) := 2 π i 1 ∫ σ 0 − i T ′ σ 0 + i T ′ ( − ζ ζ ′ ( 2 1 + i t + z ) ) k z N z d z , I := T 1 ∫ 0 T ∣ V ( t ) ∣ 2 d t . (142) We want to obtain an asymptotic expression of I k ( T , N ) I_k(T,N) I k ( T , N ) (11) . Hence, by using (135) and the definition of I k ( T , N ) I_k(T,N) I k ( T , N ) (11) , we may write
I k ( T , N ) = 1 T ∫ 0 T ∣ V ( t ) + E ( t ) ∣ 2 d t . I_k(T,N)=\frac{1}{T}\int_0^T|V(t)+E(t)|^2\,dt. I k ( T , N ) = T 1 ∫ 0 T ∣ V ( t ) + E ( t ) ∣ 2 d t . Using ∣ A + B ∣ 2 = ∣ A ∣ 2 + 2 ℜ ( A B ‾ ) + ∣ B ∣ 2 |A+B|^2=|A|^2+2\Re(A\overline{B})+|B|^2 ∣ A + B ∣ 2 = ∣ A ∣ 2 + 2ℜ ( A B ) + ∣ B ∣ 2
I k ( T , N ) − I = 2 T ℜ ∫ 0 T V ( t ) E ( t ) ‾ d t + 1 T ∫ 0 T ∣ E ( t ) ∣ 2 d t . (143) I_k(T,N)-\mathcal{I}=\frac{2}{T}\Re\int_0^T V(t)\overline{E(t)}\,dt+\frac{1}{T}\int_0^T|E(t)|^2\,dt. \tag*{(143)} I k ( T , N ) − I = T 2 ℜ ∫ 0 T V ( t ) E ( t ) d t + T 1 ∫ 0 T ∣ E ( t ) ∣ 2 d t . (143) and by the Cauchy–Schwarz inequality,
∣ I k ( T , N ) − I ∣ ≤ 2 I ( 1 T ∫ 0 T ∣ E ( t ) ∣ 2 d t ) 1 / 2 + 1 T ∫ 0 T ∣ E ( t ) ∣ 2 d t . (144) |I_k(T,N)-\mathcal{I}|\le2\sqrt{\mathcal{I}}\left(\frac{1}{T}\int_0^T|E(t)|^2\,dt\right)^{1/2}+\frac{1}{T}\int_0^T|E(t)|^2\,dt. \tag*{(144)} ∣ I k ( T , N ) − I ∣ ≤ 2 I ( T 1 ∫ 0 T ∣ E ( t ) ∣ 2 d t ) 1/2 + T 1 ∫ 0 T ∣ E ( t ) ∣ 2 d t . (144) Using ∣ E ( t ) ∣ ≪ M |E(t)| \ll\mathcal{M} ∣ E ( t ) ∣ ≪ M
I k ( T , N ) = I + O ( I M + M 2 ) . (145) \mathcal{I}_k(T,N)=\mathcal{I}+O\left(\sqrt{\mathcal{I}}\mathcal{M}+\mathcal{M}^2\right). \tag*{(145)} I k ( T , N ) = I + O ( I M + M 2 ) . (145) where
I = 1 4 π 2 T ∫ σ 0 − i T ′ σ 0 + i T ′ ∫ − σ 0 − i T ′ − σ 0 + i T ′ N z 1 − z 2 z 1 z 2 [ ∫ 0 T ( ζ ′ ζ ( 1 2 + i t + z 1 ) ) k ( ζ ′ ζ ( 1 2 − i t − z 2 ) ) k d t ] d z 2 d z 1 . (146) \mathcal{I}=\frac{1}{4\pi^2T}\int_{\sigma_0-iT'}^{\sigma_0+iT'}\int_{-\sigma_0-iT'}^{-\sigma_0+iT'}\frac{N^{z_1-z_2}}{z_1z_2}\left[\int_0^T\left(\frac{\zeta'}{\zeta}\left(\frac{1}{2}+it+z_1\right)\right)^k\left(\frac{\zeta'}{\zeta}\left(\frac{1}{2}-it-z_2\right)\right)^k\,dt\right]dz_2dz_1. \tag*{(146)} I = 4 π 2 T 1 ∫ σ 0 − i T ′ σ 0 + i T ′ ∫ − σ 0 − i T ′ − σ 0 + i T ′ z 1 z 2 N z 1 − z 2 [ ∫ 0 T ( ζ ζ ′ ( 2 1 + i t + z 1 ) ) k ( ζ ζ ′ ( 2 1 − i t − z 2 ) ) k d t ] d z 2 d z 1 . (146) It remains to bound the error term I M + M 2 \sqrt{\mathcal{I}}\mathcal{M}+\mathcal{M}^2 I M + M 2 z = σ 0 + i v z=\sigma_0+iv z = σ 0 + i v
1 2 + i t + z = 1 2 + σ 0 + i ( t + v ) , \frac{1}{2}+it+z=\frac{1}{2}+\sigma_0+i(t+v), 2 1 + i t + z = 2 1 + σ 0 + i ( t + v ) , with ∣ t + v ∣ ≤ T + T ′ ≪ T ′ |t+v|\le T+T'\ll T' ∣ t + v ∣ ≤ T + T ′ ≪ T ′ ∣ t + v ∣ |t+v| ∣ t + v ∣ (136) gives
ζ ′ ζ ( 1 2 + σ 0 + i ( t + v ) ) ≪ η , α , ε ( log T ′ ) 2 . (147) \frac{\zeta'}{\zeta}\left(\frac{1}{2}+\sigma_0+i(t+v)\right)\ll_{\eta,\alpha,\varepsilon}(\log T')^2. \tag*{(147)} ζ ζ ′ ( 2 1 + σ 0 + i ( t + v ) ) ≪ η , α , ε ( log T ′ ) 2 . (147) When ∣ t + v ∣ |t+v| ∣ t + v ∣
ζ ′ ζ ( 1 2 + σ 0 + i ( t + v ) ) ≪ σ 0 − 1 ≪ η log T . \frac{\zeta'}{\zeta}\left(\frac{1}{2}+\sigma_0+i(t+v)\right)\ll\sigma_0^{-1}\ll_{\eta}\log T. ζ ζ ′ ( 2 1 + σ 0 + i ( t + v ) ) ≪ σ 0 − 1 ≪ η log T . Therefore, the first estimate (147) holds uniformly for 0 ≤ t ≤ T 0\le t\le T 0 ≤ t ≤ T ∣ v ∣ ≤ T ′ |v|\le T' ∣ v ∣ ≤ T ′
N σ 0 = exp ( η log N log T ) = e α η + o ( 1 ) = O α , η ( 1 ) N^{\sigma_0}=\exp\left(\frac{\eta\log N}{\log T}\right)=e^{\alpha\eta+o(1)}=O_{\alpha,\eta}(1) N σ 0 = exp ( log T η log N ) = e α η + o ( 1 ) = O α , η ( 1 ) gives
∣ V ( t ) ∣ ≪ ( log T ′ ) 2 k ∫ − T ′ T ′ d v ∣ σ 0 + i v ∣ ≪ ( log T ′ ) 2 k log T . (148) |V(t)|\ll(\log T')^{2k}\int_{-T'}^{T'}\frac{dv}{|\sigma_0+iv|}\ll(\log T')^{2k}\log T. \tag*{(148)} ∣ V ( t ) ∣ ≪ ( log T ′ ) 2 k ∫ − T ′ T ′ ∣ σ 0 + i v ∣ d v ≪ ( log T ′ ) 2 k log T . (148) Hence I ≪ ( log T ) 4 k + 2 \mathcal{I}\ll(\log T)^{4k+2} I ≪ ( log T ) 4 k + 2 M ≪ ( log T ) 2 k T − δ \mathcal{M}\ll(\log T)^{2k}T^{-\delta} M ≪ ( log T ) 2 k T − δ I M + M 2 = o ( 1 ) \sqrt{\mathcal{I}}\mathcal{M}+\mathcal{M}^2=o(1) I M + M 2 = o ( 1 ) T → ∞ T\to\infty T → ∞ Conjecture 1 may be applied to the innermost integral in I \mathcal{I} I Y = 2 T ′ Y=2T' Y = 2 T ′ σ 0 = η / log T \sigma_0=\eta/\log T σ 0 = η / log T 0 < η < 3 / 2 0<\eta<3/2 0 < η < 3/2 ℜ z 1 = σ 0 \Re z_1=\sigma_0 ℜ z 1 = σ 0 ℜ z 2 = − σ 0 \Re z_2=-\sigma_0 ℜ z 2 = − σ 0
0 < ℜ z 1 , − ℜ z 2 , ℜ ( z 1 − z 2 ) ≤ 3 log T . 0<\Re z_1,-\Re z_2,\Re(z_1-z_2)\le\frac{3}{\log T}. 0 < ℜ z 1 , − ℜ z 2 , ℜ ( z 1 − z 2 ) ≤ log T 3 . Applying Conjecture 1 gives the error term
O ( ( log T ) 2 k − 1 + ξ ( 1 + min ( ∣ z 1 − z 2 ∣ , 1 ) log T ) − θ k , c ) O\left((\log T)^{2k-1+\xi}\left(1+\min(|z_1-z_2|,1)\log T\right)^{-\theta_{k,c}}\right) O ( ( log T ) 2 k − 1 + ξ ( 1 + min ( ∣ z 1 − z 2 ∣ , 1 ) log T ) − θ k , c ) for the normalised innermost integral with a fixed ξ > 0 \xi>0 ξ > 0 k = 1 , 2 k=1,2 k = 1 , 2 θ k , c = 1 + c \theta_{k,c}=1+c θ k , c = 1 + c 0 < ξ < c 0<\xi<c 0 < ξ < c k ≥ 3 k\ge3 k ≥ 3 0 < ξ < 1 0<\xi<1 0 < ξ < 1 ξ \xi ξ c c c
∣ N z 1 − z 2 ∣ = N 2 σ 0 = exp ( 2 η log N log T ) ≪ 1. |N^{z_1-z_2}|=N^{2\sigma_0}=\exp\left(\frac{2\eta\log N}{\log T}\right)\ll1. ∣ N z 1 − z 2 ∣ = N 2 σ 0 = exp ( log T 2 η log N ) ≪ 1. Thus the contribution of the conjectural error term, which we will denote by I e \mathcal{I}_e I e N z 1 − z 2 N^{z_1-z_2} N z 1 − z 2 (146) . It follows that
I e ≪ ( log T ) 2 k − 1 + ξ ∫ σ 0 − i T ′ σ 0 + i T ′ ∫ − σ 0 − i T ′ − σ 0 + i T ′ ( 1 + min ( ∣ z 1 − z 2 ∣ , 1 ) log T ) − θ k , c ∣ z 1 ∣ ∣ z 2 ∣ ∣ d z 2 ∣ ∣ d z 1 ∣ . (149) \mathcal{I}_e\ll(\log T)^{2k-1+\xi}\int_{\sigma_0-iT'}^{\sigma_0+iT'}\int_{-\sigma_0-iT'}^{-\sigma_0+iT'}\frac{\left(1+\min(|z_1-z_2|,1)\log T\right)^{-\theta_{k,c}}}{|z_1||z_2|}\,|dz_2|\,|dz_1|. \tag*{(149)} I e ≪ ( log T ) 2 k − 1 + ξ ∫ σ 0 − i T ′ σ 0 + i T ′ ∫ − σ 0 − i T ′ − σ 0 + i T ′ ∣ z 1 ∣∣ z 2 ∣ ( 1 + min ( ∣ z 1 − z 2 ∣ , 1 ) log T ) − θ k , c ∣ d z 2 ∣ ∣ d z 1 ∣. (149) Here ∣ d z j ∣ |dz_j| ∣ d z j ∣ d z j dz_j d z j z 1 = σ 0 + i t 1 z_1=\sigma_0+it_1 z 1 = σ 0 + i t 1 z 2 = − σ 0 + i t 2 z_2=-\sigma_0+it_2 z 2 = − σ 0 + i t 2
∣ z 1 ∣ − 1 ≪ 1 σ 0 + ∣ t 1 ∣ , ∣ z 2 ∣ − 1 ≪ 1 σ 0 + ∣ t 2 ∣ . |z_1|^{-1}\ll\frac{1}{\sigma_0+|t_1|},\qquad|z_2|^{-1}\ll\frac{1}{\sigma_0+|t_2|}. ∣ z 1 ∣ − 1 ≪ σ 0 + ∣ t 1 ∣ 1 , ∣ z 2 ∣ − 1 ≪ σ 0 + ∣ t 2 ∣ 1 . We may also write
1 + min ( ∣ z 1 − z 2 ∣ , 1 ) log T ≍ 1 + min ( ∣ t 1 − t 2 ∣ , 1 ) log T 1+\min(|z_1-z_2|,1)\log T \asymp1+\min(|t_1-t_2|,1)\log T 1 + min ( ∣ z 1 − z 2 ∣ , 1 ) log T ≍ 1 + min ( ∣ t 1 − t 2 ∣ , 1 ) log T as σ 0 = η / log T \sigma_0=\eta/\log T σ 0 = η / log T
I e ≪ ( log T ) 2 k − 1 + ξ ∫ − T ′ T ′ ∫ − T ′ T ′ ( 1 + min ( ∣ t 1 − t 2 ∣ , 1 ) log T ) − θ k , c ( σ 0 + ∣ t 1 ∣ ) ( σ 0 + ∣ t 2 ∣ ) d t 2 d t 1 . (150) I_e\ll(\log T)^{2k-1+\xi}\int_{-T'}^{T'}\int_{-T'}^{T'}\frac{\left(1+\min(|t_1-t_2|,1)\log T\right)^{-\theta_{k,c}}}{(\sigma_0+|t_1|)(\sigma_0+|t_2|)}\,dt_2\,dt_1. \tag*{(150)} I e ≪ ( log T ) 2 k − 1 + ξ ∫ − T ′ T ′ ∫ − T ′ T ′ ( σ 0 + ∣ t 1 ∣ ) ( σ 0 + ∣ t 2 ∣ ) ( 1 + min ( ∣ t 1 − t 2 ∣ , 1 ) log T ) − θ k , c d t 2 d t 1 . (150) To estimate this double integral, we define
f ( t ) : = 1 σ 0 + ∣ t ∣ , f(t):=\frac{1}{\sigma_0+|t|}, f ( t ) := σ 0 + ∣ t ∣ 1 , and let u = t 1 − t 2 u=t_1-t_2 u = t 1 − t 2 u u u t = t 1 t=t_1 t = t 1
t ∈ [ − T ′ , T ′ ] ∩ [ u − T ′ , u + T ′ ] . t\in[-T',T']\cap[u-T',u+T']. t ∈ [ − T ′ , T ′ ] ∩ [ u − T ′ , u + T ′ ] . Since the integrand is non-negative, we may enlarge this range to [ − T ′ , T ′ ] [-T',T'] [ − T ′ , T ′ ]
∫ − 2 T ′ 2 T ′ ( 1 + min ( ∣ u ∣ , 1 ) log T ) − θ k , c ( ∫ − T ′ T ′ f ( t ) f ( t − u ) d t ) d u . (151) \int_{-2T'}^{2T'}\left(1+\min(|u|,1)\log T\right)^{-\theta_{k,c}}\left(\int_{-T'}^{T'}f(t)f(t-u)\,dt\right)\,du. \tag*{(151)} ∫ − 2 T ′ 2 T ′ ( 1 + min ( ∣ u ∣ , 1 ) log T ) − θ k , c ( ∫ − T ′ T ′ f ( t ) f ( t − u ) d t ) d u . (151) For ∣ u ∣ ≤ σ 0 |u|\leq\sigma_0 ∣ u ∣ ≤ σ 0
∫ − T ′ T ′ f ( t ) f ( t − u ) d t ≤ ( ∫ R f ( t ) 2 d t ) 1 / 2 ( ∫ R f ( t − u ) 2 d t ) 1 / 2 = ∫ R d t ( σ 0 + ∣ t ∣ ) 2 ≪ 1 σ 0 . \begin{aligned}
\int_{-T'}^{T'}f(t)f(t-u)\,dt
&\leq\left(\int_{\mathbb{R}}f(t)^2\,dt\right)^{1/2}\left(\int_{\mathbb{R}}f(t-u)^2\,dt\right)^{1/2}\\
&=\int_{\mathbb{R}}\frac{dt}{(\sigma_0+|t|)^2}\ll\frac{1}{\sigma_0}.
\end{aligned} ∫ − T ′ T ′ f ( t ) f ( t − u ) d t ≤ ( ∫ R f ( t ) 2 d t ) 1/2 ( ∫ R f ( t − u ) 2 d t ) 1/2 = ∫ R ( σ 0 + ∣ t ∣ ) 2 d t ≪ σ 0 1 . Suppose now that ∣ u ∣ > σ 0 |u|>\sigma_0 ∣ u ∣ > σ 0 t t t ∣ t ∣ ≤ ∣ u ∣ / 2 |t|\leq|u|/2 ∣ t ∣ ≤ ∣ u ∣/2 ∣ t − u ∣ ≤ ∣ u ∣ / 2 |t-u|\leq|u|/2 ∣ t − u ∣ ≤ ∣ u ∣/2 ∣ t − u ∣ ≥ ∣ u ∣ − ∣ t ∣ ≥ ∣ u ∣ / 2 |t-u|\geq|u|-|t|\geq|u|/2 ∣ t − u ∣ ≥ ∣ u ∣ − ∣ t ∣ ≥ ∣ u ∣/2
f ( t − u ) ≪ 1 σ 0 + ∣ u ∣ . f(t-u)\ll\frac{1}{\sigma_0+|u|}. f ( t − u ) ≪ σ 0 + ∣ u ∣ 1 . It follows that
∫ ∣ t ∣ ≤ ∣ u ∣ / 2 f ( t ) f ( t − u ) d t ≪ 1 σ 0 + ∣ u ∣ ∫ ∣ t ∣ ≤ ∣ u ∣ / 2 d t σ 0 + ∣ t ∣ = 2 σ 0 + ∣ u ∣ log ( 1 + ∣ u ∣ 2 σ 0 ) ≪ 1 + log ( ∣ u ∣ / σ 0 ) σ 0 + ∣ u ∣ . \begin{aligned}
\int_{|t|\leq|u|/2}f(t)f(t-u)\,dt
&\ll\frac{1}{\sigma_0+|u|}\int_{|t|\leq|u|/2}\frac{dt}{\sigma_0+|t|}\\
&=\frac{2}{\sigma_0+|u|}\log\left(1+\frac{|u|}{2\sigma_0}\right)\\
&\ll\frac{1+\log(|u|/\sigma_0)}{\sigma_0+|u|}.
\end{aligned} ∫ ∣ t ∣ ≤ ∣ u ∣/2 f ( t ) f ( t − u ) d t ≪ σ 0 + ∣ u ∣ 1 ∫ ∣ t ∣ ≤ ∣ u ∣/2 σ 0 + ∣ t ∣ d t = σ 0 + ∣ u ∣ 2 log ( 1 + 2 σ 0 ∣ u ∣ ) ≪ σ 0 + ∣ u ∣ 1 + log ( ∣ u ∣/ σ 0 ) . The region ∣ t − u ∣ ≤ ∣ u ∣ / 2 |t-u|\leq|u|/2 ∣ t − u ∣ ≤ ∣ u ∣/2 ∣ t ∣ |t| ∣ t ∣ ∣ t − u ∣ |t-u| ∣ t − u ∣ ∣ u ∣ / 2 |u|/2 ∣ u ∣/2 ∣ t ∣ ≤ 2 ∣ u ∣ |t|\leq2|u| ∣ t ∣ ≤ 2∣ u ∣ O ( ∣ u ∣ − 2 ) O(|u|^{-2}) O ( ∣ u ∣ − 2 ) O ( ∣ u ∣ ) O(|u|) O ( ∣ u ∣ ) O ( ∣ u ∣ − 1 ) O(|u|^{-1}) O ( ∣ u ∣ − 1 ) ∣ t ∣ > 2 ∣ u ∣ |t|>2|u| ∣ t ∣ > 2∣ u ∣ ∣ t − u ∣ ≥ ∣ t ∣ − ∣ u ∣ > ∣ t ∣ / 2 |t-u|\geq|t|-|u|>|t|/2 ∣ t − u ∣ ≥ ∣ t ∣ − ∣ u ∣ > ∣ t ∣/2 O ( t − 2 ) O(t^{-2}) O ( t − 2 )
≪ ∫ ∣ t ∣ > 2 ∣ u ∣ d t t 2 ≪ 1 ∣ u ∣ ≪ 1 σ 0 + ∣ u ∣ . \ll\int_{|t|>2|u|}\frac{dt}{t^2}\ll\frac{1}{|u|}\ll\frac{1}{\sigma_0+|u|}. ≪ ∫ ∣ t ∣ > 2∣ u ∣ t 2 d t ≪ ∣ u ∣ 1 ≪ σ 0 + ∣ u ∣ 1 . Combining these estimates and using σ 0 = η / log T \sigma_0=\eta/\log T σ 0 = η / log T η \eta η
∫ − T ′ T ′ f ( t ) f ( t − u ) d t ≪ 1 + log ( 2 + ∣ u ∣ log T ) σ 0 + ∣ u ∣ . (152) \int_{-T'}^{T'}f(t)f(t-u)\,dt\ll\frac{1+\log(2+|u|\log T)}{\sigma_0+|u|}. \tag*{(152)} ∫ − T ′ T ′ f ( t ) f ( t − u ) d t ≪ σ 0 + ∣ u ∣ 1 + log ( 2 + ∣ u ∣ log T ) . (152) In order to bound (151) explicitly using the convolution bound (152) , we first split the u u u
∣ u ∣ ≤ 1 log T , 1 log T < ∣ u ∣ ≤ 1 , 1 < ∣ u ∣ ≤ 2 T ′ . |u|\leq\frac{1}{\log T},\qquad\frac{1}{\log T}<|u|\leq1,\qquad1<|u|\leq2T'. ∣ u ∣ ≤ log T 1 , log T 1 < ∣ u ∣ ≤ 1 , 1 < ∣ u ∣ ≤ 2 T ′ . The first range contributes O ( 1 ) O(1) O ( 1 )
≪ ∫ 1 / log T < ∣ u ∣ ≤ 1 ( ∣ u ∣ log T ) − θ k , c log ( 2 + ∣ u ∣ log T ) ∣ u ∣ d u . \ll\int_{1/\log T<|u|\le1} (|u|\log T)^{-\theta_{k,c}}\frac{\log(2+|u|\log T)}{|u|}\,du. ≪ ∫ 1/ l o g T < ∣ u ∣ ≤ 1 ( ∣ u ∣ log T ) − θ k , c ∣ u ∣ log ( 2 + ∣ u ∣ log T ) d u . By changing variables x = ∣ u ∣ log T x=|u|\log T x = ∣ u ∣ log T
∫ 1 log T x − θ k , c log ( 2 + x ) x d x ≪ ∫ 1 log T x − θ k , c d x ≤ ∫ 1 ∞ x − θ k , c d x = 1 θ k , c − 1 ≪ k , c 1 , \begin{aligned}
\int_1^{\log T}x^{-\theta_{k,c}}\frac{\log(2+x)}{x}\,dx\ll\int_1^{\log T}x^{-\theta_{k,c}}\,dx \\
&\le\int_1^\infty x^{-\theta_{k,c}}\,dx=\frac{1}{\theta_{k,c}-1}\ll_{k,c}1,
\end{aligned} ∫ 1 l o g T x − θ k , c x log ( 2 + x ) d x ≪ ∫ 1 l o g T x − θ k , c d x ≤ ∫ 1 ∞ x − θ k , c d x = θ k , c − 1 1 ≪ k , c 1 , as θ k , c > 1 \theta_{k,c}>1 θ k , c > 1 x − 1 log ( 2 + x ) ≪ 1 x^{-1}\log(2+x)\ll1 x − 1 log ( 2 + x ) ≪ 1 x ≥ 1 x\ge1 x ≥ 1 min ( ∣ u ∣ , 1 ) = 1 \min(|u|,1)=1 min ( ∣ u ∣ , 1 ) = 1 σ 0 + ∣ u ∣ ≍ ∣ u ∣ \sigma_0+|u|\asymp|u| σ 0 + ∣ u ∣ ≍ ∣ u ∣
≪ ( log T ) − θ k , c ∫ 1 < ∣ u ∣ ≤ 2 T ′ log ( 2 + ∣ u ∣ log T ) ∣ u ∣ d u = 2 ( log T ) − θ k , c ∫ 1 2 T ′ log ( 2 + u log T ) u d u . \ll(\log T)^{-\theta_{k,c}}\int_{1<|u|\le2T'}\frac{\log(2+|u|\log T)}{|u|}\,du=2(\log T)^{-\theta_{k,c}}\int_1^{2T'}\frac{\log(2+u\log T)}{u}\,du. ≪ ( log T ) − θ k , c ∫ 1 < ∣ u ∣ ≤ 2 T ′ ∣ u ∣ log ( 2 + ∣ u ∣ log T ) d u = 2 ( log T ) − θ k , c ∫ 1 2 T ′ u log ( 2 + u log T ) d u . As log ( 2 + u log T ) ≤ log ( 2 + 2 T ′ log T ) ≪ log T ′ \log(2+u\log T)\le\log(2+2T'\log T)\ll\log T' log ( 2 + u log T ) ≤ log ( 2 + 2 T ′ log T ) ≪ log T ′
≪ ( log T ) − θ k , c ( log T ′ ) ∫ 1 2 T ′ d u u ≪ ( log T ) − θ k , c ( log T ′ ) 2 ≪ ( log T ) 2 − θ k , c , \ll(\log T)^{-\theta_{k,c}}(\log T')\int_1^{2T'}\frac{du}{u}\ll(\log T)^{-\theta_{k,c}}(\log T')^2\ll(\log T)^{2-\theta_{k,c}}, ≪ ( log T ) − θ k , c ( log T ′ ) ∫ 1 2 T ′ u d u ≪ ( log T ) − θ k , c ( log T ′ ) 2 ≪ ( log T ) 2 − θ k , c , where the final bound uses log T ′ = γ log T \log T'=\gamma\log T log T ′ = γ log T I e \mathcal{I}_e I e ≪ 1 + ( log T ) 2 − θ k , c \ll1+(\log T)^{2-\theta_{k,c}} ≪ 1 + ( log T ) 2 − θ k , c
I e ≪ ( log T ) 2 k − 1 + ξ ( 1 + ( log T ) 2 − θ k , c ) . \mathcal{I}_e\ll(\log T)^{2k-1+\xi}\left(1+(\log T)^{2-\theta_{k,c}}\right). I e ≪ ( log T ) 2 k − 1 + ξ ( 1 + ( log T ) 2 − θ k , c ) . If k ≥ 3 k\ge3 k ≥ 3 θ k , c = k − 1 ≥ 2 \theta_{k,c}=k-1\ge2 θ k , c = k − 1 ≥ 2 I e ≪ ( log T ) 2 k − 1 + ξ = o ( ( log T ) 2 k ) \mathcal{I}_e\ll(\log T)^{2k-1+\xi}=o((\log T)^{2k}) I e ≪ ( log T ) 2 k − 1 + ξ = o (( log T ) 2 k ) 0 < ξ < 1 0<\xi<1 0 < ξ < 1 k = 1 , 2 k=1,2 k = 1 , 2 θ k , c = 1 + c \theta_{k,c}=1+c θ k , c = 1 + c
I e ≪ ( log T ) 2 k − 1 + ξ ( 1 + ( log T ) 1 − c ) ≪ ( log T ) 2 k + ξ − c , \mathcal{I}_e\ll(\log T)^{2k-1+\xi}\left(1+(\log T)^{1-c}\right)\ll(\log T)^{2k+\xi-c}, I e ≪ ( log T ) 2 k − 1 + ξ ( 1 + ( log T ) 1 − c ) ≪ ( log T ) 2 k + ξ − c , which is o ( ( log T ) 2 k ) o((\log T)^{2k}) o (( log T ) 2 k ) 0 < ξ < c 0<\xi<c 0 < ξ < c ξ \xi ξ I e = o ( ( log T ) 2 k ) \mathcal{I}_e=o((\log T)^{2k}) I e = o (( log T ) 2 k ) k ≥ 1 k\ge1 k ≥ 1
Next, we return to the leading-order term from applying Conjecture 1 with C u = C u ( z ) C_u=C_u(z) C u = C u ( z ) C v = C v ( z ) C_v=C_v(z) C v = C v ( z ) C u C_u C u 0 0 0 − z -z − z C v C_v C v 0 0 0 z z z ξ \xi ξ
I k ( T , N ) = 1 4 π 2 k ! 2 ∫ σ 0 − i T ′ σ 0 + i T ′ ∫ − σ 0 − i T ′ − σ 0 + i T ′ z − k 2 N z z 1 k z 2 k W k ( z ; T ) d z 2 d z 1 + o ( ( log T ) 2 k ) I_k(T,N)=\frac{1}{4\pi^2k!^2}\int_{\sigma_0-iT'}^{\sigma_0+iT'}\int_{-\sigma_0-iT'}^{-\sigma_0+iT'}z^{-k^2}\frac{N^z}{z_1^kz_2^k}\mathcal{W}_k(z;T)\,dz_2dz_1+o((\log T)^{2k}) I k ( T , N ) = 4 π 2 k ! 2 1 ∫ σ 0 − i T ′ σ 0 + i T ′ ∫ − σ 0 − i T ′ − σ 0 + i T ′ z − k 2 z 1 k z 2 k N z W k ( z ; T ) d z 2 d z 1 + o (( log T ) 2 k ) with
W k ( z ; T ) = ( − 1 ) k ( 2 π i ) 2 k ∫ C u k ∫ C v k T 1 2 ( ∑ j = 1 k u j − ∑ ℓ = 1 k v ℓ ) ∏ 1 ≤ i < j ≤ k ( u i − u j ) 2 ∏ 1 ≤ i < j ≤ k ( v i − v j ) 2 \mathcal{W}_k(z;T)=\frac{(-1)^k}{(2\pi i)^{2k}}\int_{C_u^k}\int_{C_v^k}T^{\frac12(\sum_{j=1}^k u_j-\sum_{\ell=1}^k v_\ell)}
\prod_{1\le i<j\le k}(u_i-u_j)^2
\prod_{1\le i<j\le k}(v_i-v_j)^2 W k ( z ; T ) = ( 2 π i ) 2 k ( − 1 ) k ∫ C u k ∫ C v k T 2 1 ( ∑ j = 1 k u j − ∑ ℓ = 1 k v ℓ ) 1 ≤ i < j ≤ k ∏ ( u i − u j ) 2 1 ≤ i < j ≤ k ∏ ( v i − v j ) 2 × ( ∑ j = 1 k 1 u j + ∑ ℓ = 1 k 1 v ℓ − z − log T 2 ) k ( ∑ j = 1 k 1 u j + z + ∑ ℓ = 1 k 1 v ℓ + log T 2 ) k {}\times\left(\sum_{j=1}^k\frac{1}{u_j}+\sum_{\ell=1}^k\frac{1}{v_\ell-z}-\frac{\log T}{2}\right)^k
\left(\sum_{j=1}^k\frac{1}{u_j+z}+\sum_{\ell=1}^k\frac{1}{v_\ell}+\frac{\log T}{2}\right)^k × ( j = 1 ∑ k u j 1 + ℓ = 1 ∑ k v ℓ − z 1 − 2 log T ) k ( j = 1 ∑ k u j + z 1 + ℓ = 1 ∑ k v ℓ 1 + 2 log T ) k × ∏ j , ℓ = 1 k ( u j − v ℓ + z ) ∏ j = 1 k u j − k d u j ∏ ℓ = 1 k v ℓ − k d v ℓ . (153) {}\times\prod_{j,\ell=1}^k(u_j-v_\ell+z)\prod_{j=1}^k u_j^{-k}\,du_j\prod_{\ell=1}^k v_\ell^{-k}\,dv_\ell. \tag*{(153)} × j , ℓ = 1 ∏ k ( u j − v ℓ + z ) j = 1 ∏ k u j − k d u j ℓ = 1 ∏ k v ℓ − k d v ℓ . (153) where z = z 1 − z 2 z=z_1-z_2 z = z 1 − z 2 ( − 1 ) k (-1)^k ( − 1 ) k ∏ j , ℓ ( v ℓ − u j − z ) \prod_{j,\ell}(v_\ell-u_j-z) ∏ j , ℓ ( v ℓ − u j − z ) ∏ j , ℓ ( u j − v ℓ + z ) \prod_{j,\ell}(u_j-v_\ell+z) ∏ j , ℓ ( u j − v ℓ + z ) z z z ℜ z = 2 σ 0 \Re z=2\sigma_0 ℜ z = 2 σ 0 Lemma 2.7 to rewrite W k ( z ; T ) \mathcal{W}_k(z;T) W k ( z ; T ) I k ( T , N ) I_k(T,N) I k ( T , N ) U = ∑ j = 1 k u j U=\sum_{j=1}^k u_j U = ∑ j = 1 k u j V = ∑ ℓ = 1 k v ℓ V=\sum_{\ell=1}^k v_\ell V = ∑ ℓ = 1 k v ℓ L : = log T L := \log T L := log T T 1 2 ( U − V ) = e L 2 ( U − V ) T^{\frac{1}{2}(U-V)} = e^{\frac{L}{2}(U-V)} T 2 1 ( U − V ) = e 2 L ( U − V ) s = L 2 s = \frac{L}{2} s = 2 L t = − L 2 t = -\frac{L}{2} t = − 2 L
W k ( z ; T ) = ( − 1 ) k ( 2 π i ) 2 k ∫ C u k ∫ C v k T ∑ j = 1 k u j Δ ( u ) 2 Δ ( v ) 2 × ( ∑ j = 1 k 1 u j + ∑ ℓ = 1 k 1 v ℓ − z − log T ) k ( ∑ j = 1 k 1 u j + z + ∑ ℓ = 1 k 1 v ℓ ) k × ∏ j , ℓ = 1 k ( u j − v ℓ + z ) ∏ j = 1 k u j − k d u j ∏ ℓ = 1 k v ℓ − k d v ℓ . (154) \begin{aligned}
W_k(z;T)={}&\frac{(-1)^k}{(2\pi i)^{2k}}\int_{\mathcal{C}_u^k}\int_{\mathcal{C}_v^k} T^{\sum_{j=1}^k u_j}\Delta(u)^2\Delta(v)^2\\
&\times\left(\sum_{j=1}^k\frac{1}{u_j}+\sum_{\ell=1}^k\frac{1}{v_\ell-z}-\log T\right)^k
\left(\sum_{j=1}^k\frac{1}{u_j+z}+\sum_{\ell=1}^k\frac{1}{v_\ell}\right)^k\\
&\times\prod_{j,\ell=1}^k(u_j-v_\ell+z)\prod_{j=1}^k u_j^{-k}\,du_j\prod_{\ell=1}^k v_\ell^{-k}\,dv_\ell. \tag*{(154)}
\end{aligned} W k ( z ; T ) = ( 2 π i ) 2 k ( − 1 ) k ∫ C u k ∫ C v k T ∑ j = 1 k u j Δ ( u ) 2 Δ ( v ) 2 × ( j = 1 ∑ k u j 1 + ℓ = 1 ∑ k v ℓ − z 1 − log T ) k ( j = 1 ∑ k u j + z 1 + ℓ = 1 ∑ k v ℓ 1 ) k × j , ℓ = 1 ∏ k ( u j − v ℓ + z ) j = 1 ∏ k u j − k d u j ℓ = 1 ∏ k v ℓ − k d v ℓ . (154) Thus, the symmetric pair of shifts ( − log T / 2 , + log T / 2 ) (-\log T/2,+\log T/2) ( − log T /2 , + log T /2 ) ( − log T , 0 ) (-\log T,0) ( − log T , 0 ) 2 k 2k 2 k W k ( z ; T ) W_k(z;T) W k ( z ; T ) Lemma 2.6 can be applied by setting σ = − z \sigma=-z σ = − z L = log T L=\log T L = log T Φ L ( u 1 , … , u k ) = T ∑ j = 1 k u j \Phi_L(u_1,\ldots,u_k)=T^{\sum_{j=1}^k u_j} Φ L ( u 1 , … , u k ) = T ∑ j = 1 k u j
I k ( T , N ) = 1 4 π 2 k ! 2 ∑ r = 0 k ( k r ) 2 ∑ a = 0 k ( k a ) L k − a × ∫ σ 0 − i T ′ σ 0 + i T ′ ∫ − σ 0 − i T ′ − σ 0 + i T ′ N z z 1 z 2 z − a − k J r ( a ) ( − z ; T ) d z 2 d z 1 + o ( ( log T ) 2 k ) . (155) \begin{aligned}
I_k(T,N)={}&\frac{1}{4\pi^2 k!^2}\sum_{r=0}^k\binom{k}{r}^2\sum_{a=0}^k\binom{k}{a}L^{k-a}\\
&\times\int_{\sigma_0-iT'}^{\sigma_0+iT'}\int_{-\sigma_0-iT'}^{-\sigma_0+iT'}\frac{N^z}{z_1z_2}z^{-a-k}J_r^{(a)}(-z;T)\,dz_2dz_1+o((\log T)^{2k}). \tag*{(155)}
\end{aligned} I k ( T , N ) = 4 π 2 k ! 2 1 r = 0 ∑ k ( r k ) 2 a = 0 ∑ k ( a k ) L k − a × ∫ σ 0 − i T ′ σ 0 + i T ′ ∫ − σ 0 − i T ′ − σ 0 + i T ′ z 1 z 2 N z z − a − k J r ( a ) ( − z ; T ) d z 2 d z 1 + o (( log T ) 2 k ) . (155) where
J r ( a ) ( − z ; T ) = 1 ( 2 π i ) 2 k ∫ C c ^ j ( r ) ∫ C c ^ ℓ ( r ) T − z ∑ j = 1 k x j Δ ( x ) 2 Δ ( y ) 2 ∏ j , ℓ = 1 k ( x j − y ℓ − 1 ) × ( ∑ j = 1 k 1 x j + ∑ ℓ = 1 k 1 y ℓ + 1 ) a ( ∑ j = 1 k 1 x j − 1 + ∑ ℓ = 1 k 1 y ℓ ) k ∏ j = 1 k x j − k d x j ∏ ℓ = 1 k y ℓ − k d y ℓ . (156) \begin{aligned}
J_r^{(a)}(-z;T)={}&\frac{1}{(2\pi i)^{2k}}\int_{\mathcal{C}_{\hat c_j(r)}}\int_{\mathcal{C}_{\hat c_\ell(r)}}T^{-z\sum_{j=1}^k x_j}\Delta(x)^2\Delta(y)^2\prod_{j,\ell=1}^k(x_j-y_\ell-1)\\
&\times\left(\sum_{j=1}^k\frac{1}{x_j}+\sum_{\ell=1}^k\frac{1}{y_\ell+1}\right)^a
\left(\sum_{j=1}^k\frac{1}{x_j-1}+\sum_{\ell=1}^k\frac{1}{y_\ell}\right)^k
\prod_{j=1}^k x_j^{-k}\,dx_j\prod_{\ell=1}^k y_\ell^{-k}\,dy_\ell. \tag*{(156)}
\end{aligned} J r ( a ) ( − z ; T ) = ( 2 π i ) 2 k 1 ∫ C c ^ j ( r ) ∫ C c ^ ℓ ( r ) T − z ∑ j = 1 k x j Δ ( x ) 2 Δ ( y ) 2 j , ℓ = 1 ∏ k ( x j − y ℓ − 1 ) × ( j = 1 ∑ k x j 1 + ℓ = 1 ∑ k y ℓ + 1 1 ) a ( j = 1 ∑ k x j − 1 1 + ℓ = 1 ∑ k y ℓ 1 ) k j = 1 ∏ k x j − k d x j ℓ = 1 ∏ k y ℓ − k d y ℓ . (156) with c j ( r ) c_j(r) c j ( r ) c ^ ℓ ( r ) \hat c_\ell(r) c ^ ℓ ( r ) Lemma 2.6 . For each fixed pair ( r , a ) (r,a) ( r , a ) z = z 1 − z 2 z=z_1-z_2 z = z 1 − z 2
− 1 ( 2 π i ) 2 ∫ σ 0 − i T ′ σ 0 + i T ′ ∫ − σ 0 − i T ′ − σ 0 + i T ′ z − a − k J r ( a ) ( − z ; T ) N z z 1 z 2 d z 2 d z 1 = − 1 2 π i ∫ 2 σ 0 − 2 i T ′ 2 σ 0 + 2 i T ′ z − a − k J r ( a ) ( − z ; T ) N z K T ′ ( z ) d z . (157) \begin{aligned}
-\frac{1}{(2\pi i)^2}\int_{\sigma_0-iT'}^{\sigma_0+iT'}\int_{-\sigma_0-iT'}^{-\sigma_0+iT'}z^{-a-k}J_r^{(a)}(-z;T)\frac{N^z}{z_1z_2}\,dz_2dz_1\\
=-\frac{1}{2\pi i}\int_{2\sigma_0-2iT'}^{2\sigma_0+2iT'}z^{-a-k}J_r^{(a)}(-z;T)N^zK_{T'}(z)\,dz. \tag*{(157)}
\end{aligned} − ( 2 π i ) 2 1 ∫ σ 0 − i T ′ σ 0 + i T ′ ∫ − σ 0 − i T ′ − σ 0 + i T ′ z − a − k J r ( a ) ( − z ; T ) z 1 z 2 N z d z 2 d z 1 = − 2 π i 1 ∫ 2 σ 0 − 2 i T ′ 2 σ 0 + 2 i T ′ z − a − k J r ( a ) ( − z ; T ) N z K T ′ ( z ) d z . (157) where
K T ′ ( z ) : = − 1 2 π i ∫ − σ 0 + i max ( − T ′ , − T ′ − ℑ z ) − σ 0 + i min ( T ′ , T ′ − ℑ z ) d z 2 ( z + z 2 ) z 2 = 1 z + O ( log T T ′ ) . (158) K_{T'}(z):=-\frac{1}{2\pi i}\int_{-\sigma_0+i\max(-T',-T'-\Im z)}^{-\sigma_0+i\min(T',T'-\Im z)}\frac{dz_2}{(z+z_2)z_2}=\frac{1}{z}+O\left(\frac{\log T}{T'}\right). \tag*{(158)} K T ′ ( z ) := − 2 π i 1 ∫ − σ 0 + i m a x ( − T ′ , − T ′ − ℑ z ) − σ 0 + i m i n ( T ′ , T ′ − ℑ z ) ( z + z 2 ) z 2 d z 2 = z 1 + O ( T ′ log T ) . (158) uniformly for z = 2 σ 0 + i y z=2\sigma_0+iy z = 2 σ 0 + i y ∣ y ∣ ≤ 2 T ′ |y|\leq2T' ∣ y ∣ ≤ 2 T ′ (158) by considering two ranges; suppose first that ∣ y ∣ ≤ T ′ / 2 |y|\leq T'/2 ∣ y ∣ ≤ T ′ /2 z 2 z_2 z 2 z 2 = − z z_2=-z z 2 = − z z 2 = 0 z_2=0 z 2 = 0
Res z 2 = − z 1 ( z + z 2 ) z 2 = − 1 z , \operatorname{Res}_{z_2=-z}\frac{1}{(z+z_2)z_2}=-\frac{1}{z}, Res z 2 = − z ( z + z 2 ) z 2 1 = − z 1 , so the minus sign in the definition of K T ′ ( z ) K_{T'}(z) K T ′ ( z ) 1 / z 1/z 1/ z z 2 = x + i v z_2=x+iv z 2 = x + i v x ≤ − σ 0 x\leq-\sigma_0 x ≤ − σ 0 ∣ y ∣ ≤ T ′ / 2 |y|\leq T'/2 ∣ y ∣ ≤ T ′ /2
∣ z 2 ∣ ≫ T ′ + ∣ x ∣ , ∣ z + z 2 ∣ ≫ T ′ + ∣ x ∣ . |z_2|\gg T'+|x|,\qquad|z+z_2|\gg T'+|x|. ∣ z 2 ∣ ≫ T ′ + ∣ x ∣ , ∣ z + z 2 ∣ ≫ T ′ + ∣ x ∣. Their total contribution is therefore
≪ ∫ − ∞ − σ 0 d x ( T ′ + ∣ x ∣ ) 2 ≪ 1 T ′ , \ll\int_{-\infty}^{-\sigma_0} \frac{dx}{(T' + |x|)^2} \ll\frac{1}{T'}, ≪ ∫ − ∞ − σ 0 ( T ′ + ∣ x ∣ ) 2 d x ≪ T ′ 1 , and this gives
K T ′ ( z ) = 1 z + O ( 1 T ′ ) , ∣ y ∣ ≤ T ′ 2 . (159) K_{T'}(z) = \frac{1}{z} + O\left(\frac{1}{T'}\right), \qquad|y| \le\frac{T'}{2}. \tag*{(159)} K T ′ ( z ) = z 1 + O ( T ′ 1 ) , ∣ y ∣ ≤ 2 T ′ . (159) Now we consider the case T ′ / 2 < ∣ y ∣ ≤ 2 T ′ T'/2 < |y| \le2T' T ′ /2 < ∣ y ∣ ≤ 2 T ′ z 2 = − σ 0 + i v z_2 = -\sigma_0 + iv z 2 = − σ 0 + i v (152) with u = − y u=-y u = − y
∣ K T ′ ( z ) ∣ ≪ ∫ − T ′ T ′ d v ( σ 0 + ∣ v ∣ ) ( σ 0 + ∣ v + y ∣ ) ≪ 1 + log ( 2 + ∣ y ∣ log T ) σ 0 + ∣ y ∣ ≪ log T T ′ . |K_{T'}(z)| \ll\int_{-T'}^{T'} \frac{dv}{(\sigma_0+|v|)(\sigma_0+|v+y|)} \ll\frac{1+\log(2+|y|\log T)}{\sigma_0+|y|} \ll\frac{\log T}{T'}. ∣ K T ′ ( z ) ∣ ≪ ∫ − T ′ T ′ ( σ 0 + ∣ v ∣ ) ( σ 0 + ∣ v + y ∣ ) d v ≪ σ 0 + ∣ y ∣ 1 + log ( 2 + ∣ y ∣ log T ) ≪ T ′ log T . Here the last estimate follows from T ′ / 2 < ∣ y ∣ ≤ 2 T ′ T'/2 < |y| \le2T' T ′ /2 < ∣ y ∣ ≤ 2 T ′ T ′ = T γ T'=T^\gamma T ′ = T γ ∣ z ∣ ≥ ∣ y ∣ > T ′ / 2 |z| \ge|y| > T'/2 ∣ z ∣ ≥ ∣ y ∣ > T ′ /2 ∣ z ∣ − 1 ≪ T ′ − 1 |z|^{-1} \ll T'^{-1} ∣ z ∣ − 1 ≪ T ′ − 1
∣ K T ′ ( z ) − 1 z ∣ ≤ ∣ K T ′ ( z ) ∣ + 1 ∣ z ∣ ≪ log T T ′ . (160) \left|K_{T'}(z)-\frac{1}{z}\right| \le|K_{T'}(z)|+\frac{1}{|z|} \ll\frac{\log T}{T'}. \tag*{(160)} K T ′ ( z ) − z 1 ≤ ∣ K T ′ ( z ) ∣ + ∣ z ∣ 1 ≪ T ′ log T . (160) Combining (159) and (160) proves (158) uniformly for ∣ y ∣ ≤ 2 T ′ |y|\le2T' ∣ y ∣ ≤ 2 T ′
The above kernel estimate reduces the remaining task to controlling the z z z J r ( a ) ( − z ; T ) J_r^{(a)}(-z;T) J r ( a ) ( − z ; T )
∑ j = 1 k x j = r + ∑ j = 1 r ( x j − 1 ) + ∑ j = r + 1 k x j , (161) \sum_{j=1}^{k}x_j=r+\sum_{j=1}^{r}(x_j-1)+\sum_{j=r+1}^{k}x_j, \tag*{(161)} j = 1 ∑ k x j = r + j = 1 ∑ r ( x j − 1 ) + j = r + 1 ∑ k x j , (161) we have the factorisation
J r ( a ) ( − z ; T ) = T − r z J ~ r ( a ) ( z ; T ) (162) J_r^{(a)}(-z;T)=T^{-rz}\widetilde{J}_r^{(a)}(z;T) \tag*{(162)} J r ( a ) ( − z ; T ) = T − r z J r ( a ) ( z ; T ) (162) with
J ~ r ( a ) ( z ; T ) : = 1 ( 2 π i ) 2 k ∫ c j ( r ) ∫ c ^ ℓ ( r ) exp ( − z log T { ∑ j = 1 r ( x j − 1 ) + ∑ j = r + 1 k x j } ) × ( ∑ j = 1 k 1 x j + ∑ ℓ = 1 k 1 y ℓ + 1 ) ( ∑ j = 1 k 1 x j − 1 + ∑ ℓ = 1 k 1 y ℓ ) a ∏ j , ℓ = 1 k ( x j − y ℓ − 1 ) × Δ ( x ) 2 Δ ( y ) 2 ∏ j = 1 k x j − k d x j ∏ ℓ = 1 k y ℓ − k d y ℓ . (163) \begin{aligned}
\widetilde{J}_r^{(a)}(z;T) &:=
\frac{1}{(2\pi i)^{2k}}
\int_{c_j(r)}\int_{\widehat{c}_\ell(r)}
\exp\left(-z\log T\left\{\sum_{j=1}^{r}(x_j-1)+\sum_{j=r+1}^{k}x_j\right\}\right)\\
&\quad\times\left(\sum_{j=1}^{k}\frac{1}{x_j}+\sum_{\ell=1}^{k}\frac{1}{y_\ell+1}\right)
\left(\sum_{j=1}^{k}\frac{1}{x_j-1}+\sum_{\ell=1}^{k}\frac{1}{y_\ell}\right)^a
\prod_{j,\ell=1}^{k}(x_j-y_\ell-1)\\
&\quad\times\Delta(x)^2\Delta(y)^2\prod_{j=1}^{k}x_j^{-k}\,dx_j\prod_{\ell=1}^{k}y_\ell^{-k}\,dy_\ell. \tag*{(163)}
\end{aligned} J r ( a ) ( z ; T ) := ( 2 π i ) 2 k 1 ∫ c j ( r ) ∫ c ℓ ( r ) exp ( − z log T { j = 1 ∑ r ( x j − 1 ) + j = r + 1 ∑ k x j } ) × ( j = 1 ∑ k x j 1 + ℓ = 1 ∑ k y ℓ + 1 1 ) ( j = 1 ∑ k x j − 1 1 + ℓ = 1 ∑ k y ℓ 1 ) a j , ℓ = 1 ∏ k ( x j − y ℓ − 1 ) × Δ ( x ) 2 Δ ( y ) 2 j = 1 ∏ k x j − k d x j ℓ = 1 ∏ k y ℓ − k d y ℓ . (163) We now bound the z z z J ~ r ( a ) ( z ; T ) \widetilde{J}_r^{(a)}(z;T) J r ( a ) ( z ; T )
exp ( − z log T { ∑ j = 1 r ( x j − 1 ) + ∑ j = r + 1 k x j } ) = ∑ β ∈ Z ≥ 0 r ∑ γ ∈ Z ≥ 0 k − r ( − z log T ) ∣ β ∣ + ∣ γ ∣ β ! γ ! × ∏ j = 1 r ( x j − 1 ) β j ∏ j = r + 1 k x j γ j − r . (164) \begin{aligned}
\exp\left(-z\log T\left\{\sum_{j=1}^{r}(x_j-1)+\sum_{j=r+1}^{k}x_j\right\}\right)
&=\sum_{\beta\in\mathbb{Z}_{\geq0}^{r}}\sum_{\gamma\in\mathbb{Z}_{\geq0}^{k-r}}
\frac{(-z\log T)^{|\beta|+|\gamma|}}{\beta!\gamma!}\\
&\quad\times\prod_{j=1}^{r}(x_j-1)^{\beta_j}\prod_{j=r+1}^{k}x_j^{\gamma_{j-r}}. \tag*{(164)}
\end{aligned} exp ( − z log T { j = 1 ∑ r ( x j − 1 ) + j = r + 1 ∑ k x j } ) = β ∈ Z ≥ 0 r ∑ γ ∈ Z ≥ 0 k − r ∑ β ! γ ! ( − z log T ) ∣ β ∣ + ∣ γ ∣ × j = 1 ∏ r ( x j − 1 ) β j j = r + 1 ∏ k x j γ j − r . (164) This is precisely the Taylor expansion appearing in Lemma 2.8, with σ = − z \sigma=-z σ = − z Φ log T ( u 1 , … , u k ) = T ∑ j u j \Phi_{\log T}(u_1,\ldots,u_k)=T^{\sum_j u_j} Φ l o g T ( u 1 , … , u k ) = T ∑ j u j T − r z T^{-rz} T − r z a < r a<r a < r
J ~ r ( a ) ( z ; T ) = 0. (165) \widetilde{J}_r^{(a)}(z;T)=0. \tag*{(165)} J r ( a ) ( z ; T ) = 0. (165) If a ≥ r a \ge r a ≥ r (164) to the terms satisfying
∣ β ∣ ≤ k − r , ∣ γ ∣ ≤ a − r . |\beta| \le k-r,\qquad|\gamma| \le a-r. ∣ β ∣ ≤ k − r , ∣ γ ∣ ≤ a − r . Consequently, for 1 ≤ r ≤ k 1 \le r \le k 1 ≤ r ≤ k
( z log T ) m , 0 ≤ m ≤ ( k − r ) + ( a − r ) = k + a − 2 r , (z\log T)^m,\qquad0 \le m \le(k-r)+(a-r)=k+a-2r, ( z log T ) m , 0 ≤ m ≤ ( k − r ) + ( a − r ) = k + a − 2 r , can contribute to J ~ r ( a ) ( z ; T ) \widetilde{J}_{r}^{(a)}(z;T) J r ( a ) ( z ; T )
J ~ r ( a ) ( z ; T ) ≪ k ( 1 + ∣ z ∣ log T ) k + a − 2 r , 1 ≤ r ≤ k , a ≥ r . (166) \widetilde{J}_{r}^{(a)}(z;T) \ll_k (1+|z|\log T)^{k+a-2r},\qquad1\le r\le k,\quad a\ge r. \tag*{(166)} J r ( a ) ( z ; T ) ≪ k ( 1 + ∣ z ∣ log T ) k + a − 2 r , 1 ≤ r ≤ k , a ≥ r . (166) For r = 0 r=0 r = 0 (164) is empty and Lemma 2.8 leaves only the restriction ∣ γ ∣ ≤ a |\gamma|\le a ∣ γ ∣ ≤ a
J ~ 0 ( a ) ( z ; T ) ≪ k ( 1 + ∣ z ∣ log T ) a . (167) \widetilde{J}_{0}^{(a)}(z;T)\ll_k (1+|z|\log T)^a. \tag*{(167)} J 0 ( a ) ( z ; T ) ≪ k ( 1 + ∣ z ∣ log T ) a . (167) Thus
J ~ r ( a ) ( z ; T ) = 0 ( a < r ) , (168) \widetilde{J}_{r}^{(a)}(z;T)=0\qquad(a<r), \tag*{(168)} J r ( a ) ( z ; T ) = 0 ( a < r ) , (168) and, for a ≥ r a\ge r a ≥ r
J ~ r ( a ) ( z ; T ) = { O k ( ( 1 + ∣ z ∣ log T ) a ) , r = 0 , O k ( ( 1 + ∣ z ∣ log T ) k + a − 2 r ) , 1 ≤ r ≤ k . (169) \widetilde{J}_{r}^{(a)}(z;T)=
\begin{cases}
O_k\left((1+|z|\log T)^a\right), & r=0,\\
O_k\left((1+|z|\log T)^{k+a-2r}\right), & 1\le r\le k.
\end{cases} \tag*{(169)} J r ( a ) ( z ; T ) = { O k ( ( 1 + ∣ z ∣ log T ) a ) , O k ( ( 1 + ∣ z ∣ log T ) k + a − 2 r ) , r = 0 , 1 ≤ r ≤ k . (169) We may now return to the kernel K T ′ K_{T'} K T ′ (158) . For a ≥ r a\ge r a ≥ r ( r , a ) (r,a) ( r , a ) I k ( T , N ) I_k(T,N) I k ( T , N )
1 2 π i ∫ 2 σ 0 − 2 i T ′ 2 σ 0 + 2 i T ′ ( log T ) k − a ( N T r ) z z − a − k J ~ r ( a ) ( z ; T ) K T ′ ( z ) d z . (170) \frac{1}{2\pi i}\int_{2\sigma_0-2iT'}^{2\sigma_0+2iT'}(\log T)^{k-a}\left(\frac{N}{T^r}\right)^z z^{-a-k}\widetilde{J}_{r}^{(a)}(z;T)K_{T'}(z)\,dz. \tag*{(170)} 2 π i 1 ∫ 2 σ 0 − 2 i T ′ 2 σ 0 + 2 i T ′ ( log T ) k − a ( T r N ) z z − a − k J r ( a ) ( z ; T ) K T ′ ( z ) d z . (170) The estimate (158) is independent of a a a r r r
∣ K T ′ ( z ) − 1 z ∣ ≪ log T T ′ \left|K_{T'}(z)-\frac{1}{z}\right|\ll\frac{\log T}{T'} K T ′ ( z ) − z 1 ≪ T ′ log T uniformly for z = 2 σ 0 + i y z=2\sigma_0+iy z = 2 σ 0 + i y ∣ y ∣ ≤ 2 T ′ |y|\le2T' ∣ y ∣ ≤ 2 T ′ a < r a<r a < r (165) , so it remains only to consider a ≥ r a\ge r a ≥ r J ~ r ( a ) ( z ; T ) \widetilde{J}_{r}^{(a)}(z;T) J r ( a ) ( z ; T ) ( r , a ) (r,a) ( r , a ) L 1 L^1 L 1
∫ − 2 T ′ 2 T ′ ( log T ) k − a ∣ z ∣ − a − k ∣ J ~ r ( a ) ( z ; T ) ∣ d y ≪ k ( log T ) 2 k − 1 , z = 2 σ 0 + i y , (171) \int_{-2T'}^{2T'}(\log T)^{k-a}|z|^{-a-k}\left|\widetilde{J}_{r}^{(a)}(z;T)\right|\,dy\ll_k(\log T)^{2k-1},\qquad z=2\sigma_0+iy, \tag*{(171)} ∫ − 2 T ′ 2 T ′ ( log T ) k − a ∣ z ∣ − a − k J r ( a ) ( z ; T ) d y ≪ k ( log T ) 2 k − 1 , z = 2 σ 0 + i y , (171) uniformly in r , a r,a r , a
∣ ( N T r ) z ∣ = exp ( 2 σ 0 ( log N − r log T ) ) ≪ α , η , k 1 , \left|\left(\frac{N}{T^r}\right)^z\right|=\exp(2\sigma_0(\log N-r\log T))\ll_{\alpha,\eta,k}1, ( T r N ) z = exp ( 2 σ 0 ( log N − r log T )) ≪ α , η , k 1 , since σ 0 = η / log T \sigma_0=\eta/\log T σ 0 = η / log T log N = α log T + o ( log T ) \log N=\alpha\log T+o(\log T) log N = α log T + o ( log T ) (171) , we also use (169) and ∣ z ∣ ≍ σ 0 + ∣ y ∣ |z|\asymp\sigma_0+|y| ∣ z ∣ ≍ σ 0 + ∣ y ∣ r = 0 r=0 r = 0
( log T ) k − a ∣ z ∣ − a − k ∣ J ~ 0 ( a ) ( z ; T ) ∣ ≪ k ( log T ) k − a ∣ z ∣ − a − k ( 1 + ∣ z ∣ log T ) a ≪ k ( log T ) k ( σ 0 + ∣ y ∣ ) − k . (\log T)^{k-a}|z|^{-a-k}\left|\widetilde{J}_{0}^{(a)}(z;T)\right|
\ll_k(\log T)^{k-a}|z|^{-a-k}(1+|z|\log T)^a
\ll_k(\log T)^k(\sigma_0+|y|)^{-k}. ( log T ) k − a ∣ z ∣ − a − k J 0 ( a ) ( z ; T ) ≪ k ( log T ) k − a ∣ z ∣ − a − k ( 1 + ∣ z ∣ log T ) a ≪ k ( log T ) k ( σ 0 + ∣ y ∣ ) − k . For k ≥ 2 k\ge2 k ≥ 2
∫ − 2 T ′ 2 T ′ ( log T ) k ( σ 0 + ∣ y ∣ ) − k d y ≪ k ( log T ) k σ 0 1 − k ≪ ( log T ) 2 k − 1 . \int_{-2T'}^{2T'}(\log T)^k(\sigma_0+|y|)^{-k}\,dy
\ll_k(\log T)^k\sigma_0^{1-k}\ll(\log T)^{2k-1}. ∫ − 2 T ′ 2 T ′ ( log T ) k ( σ 0 + ∣ y ∣ ) − k d y ≪ k ( log T ) k σ 0 1 − k ≪ ( log T ) 2 k − 1 . For k = 1 k=1 k = 1 (163) gives
J ~ 0 ( 0 ) ( z ; T ) = 0 , J ~ 0 ( 1 ) ( z ; T ) = 1. \widetilde{J}_{0}^{(0)}(z;T)=0,\qquad\widetilde{J}_{0}^{(1)}(z;T)=1. J 0 ( 0 ) ( z ; T ) = 0 , J 0 ( 1 ) ( z ; T ) = 1. Consequently, for a = 0 , 1 a=0,1 a = 0 , 1 z = 2 σ 0 + i y z=2\sigma_0+iy z = 2 σ 0 + i y
∫ − 2 T ′ 2 T ′ ( log T ) 1 − a ∣ z ∣ − a − 1 ∣ J ~ 0 ( a ) ( z ; T ) ∣ d y ≪ ∫ − 2 T ′ 2 T ′ d y 4 σ 0 2 + y 2 ≪ σ 0 − 1 ≪ log T . \int_{-2T'}^{2T'}(\log T)^{1-a}|z|^{-a-1}|\widetilde{J}^{(a)}_0(z;T)|\,dy
\ll\int_{-2T'}^{2T'}\frac{dy}{4\sigma_0^2+y^2}\ll\sigma_0^{-1}\ll\log T. ∫ − 2 T ′ 2 T ′ ( log T ) 1 − a ∣ z ∣ − a − 1 ∣ J 0 ( a ) ( z ; T ) ∣ d y ≪ ∫ − 2 T ′ 2 T ′ 4 σ 0 2 + y 2 d y ≪ σ 0 − 1 ≪ log T . Thus the r = 0 r=0 r = 0 (171) for every k ≥ 1 k\ge1 k ≥ 1
Similarly, for 1 ≤ r ≤ k 1\le r\le k 1 ≤ r ≤ k a ≥ r a\ge r a ≥ r
( log T ) k − a ∣ z ∣ − a − k ∣ J ~ r ( a ) ( z ; T ) ∣ ≪ k ( log T ) k − a ∣ z ∣ − a − k ( 1 + ∣ z ∣ log T ) k + a − 2 r ≪ k ( log T ) 2 k − 2 r ( σ 0 + ∣ y ∣ ) − 2 r , (\log T)^{k-a}|z|^{-a-k}|\widetilde{J}^{(a)}_r(z;T)|
\ll_k(\log T)^{k-a}|z|^{-a-k}(1+|z|\log T)^{k+a-2r}
\ll_k(\log T)^{2k-2r}(\sigma_0+|y|)^{-2r}, ( log T ) k − a ∣ z ∣ − a − k ∣ J r ( a ) ( z ; T ) ∣ ≪ k ( log T ) k − a ∣ z ∣ − a − k ( 1 + ∣ z ∣ log T ) k + a − 2 r ≪ k ( log T ) 2 k − 2 r ( σ 0 + ∣ y ∣ ) − 2 r , and this leads to
∫ − 2 T ′ 2 T ′ ( log T ) 2 k − 2 r ( σ 0 + ∣ y ∣ ) − 2 r d y ≪ k ( log T ) 2 k − 2 r σ 0 1 − 2 r ≪ ( log T ) 2 k − 1 \int_{-2T'}^{2T'}(\log T)^{2k-2r}(\sigma_0+|y|)^{-2r}\,dy
\ll_k(\log T)^{2k-2r}\sigma_0^{1-2r}\ll(\log T)^{2k-1} ∫ − 2 T ′ 2 T ′ ( log T ) 2 k − 2 r ( σ 0 + ∣ y ∣ ) − 2 r d y ≪ k ( log T ) 2 k − 2 r σ 0 1 − 2 r ≪ ( log T ) 2 k − 1 for 1 ≤ r ≤ k 1\le r\le k 1 ≤ r ≤ k (171) . Therefore the contribution of the error term K T ′ ( z ) − 1 / z K_{T'}(z)-1/z K T ′ ( z ) − 1/ z ( r , a ) (r,a) ( r , a )
≪ k ( log T ) 2 k + 1 T ′ = o ( ( log T ) 2 k ) , \ll_k\frac{(\log T)^{2k+1}}{T'}=o((\log T)^{2k}), ≪ k T ′ ( log T ) 2 k + 1 = o (( log T ) 2 k ) , and the same bound holds for the total error across the sums over r r r a a a K T ′ ( z ) K_{T'}(z) K T ′ ( z ) 1 / z 1/z 1/ z (170) with an error o ( ( log T ) 2 k ) o((\log T)^{2k}) o (( log T ) 2 k )
I k ( T , N ) = 1 k ! 2 ∑ r = 0 k ( k r ) 2 ∑ a = r k ( k a ) 1 2 π i ∫ 2 σ 0 − 2 i T ′ 2 σ 0 + 2 i T ′ ( log T ) k − a ( N T r ) z z − a − k − 1 J ~ r ( a ) ( z ; T ) d z + o ( ( log T ) 2 k ) . (172) I_k(T,N)=\frac{1}{k!^2}\sum_{r=0}^{k}\binom{k}{r}^2\sum_{a=r}^{k}\binom{k}{a}\frac{1}{2\pi i}\int_{2\sigma_0-2iT'}^{2\sigma_0+2iT'}(\log T)^{k-a}\left(\frac{N}{T^r}\right)^z z^{-a-k-1}\widetilde{J}^{(a)}_r(z;T)\,dz+o((\log T)^{2k}). \tag*{(172)} I k ( T , N ) = k ! 2 1 r = 0 ∑ k ( r k ) 2 a = r ∑ k ( a k ) 2 π i 1 ∫ 2 σ 0 − 2 i T ′ 2 σ 0 + 2 i T ′ ( log T ) k − a ( T r N ) z z − a − k − 1 J r ( a ) ( z ; T ) d z + o (( log T ) 2 k ) . (172) Here the lower limit a = r a=r a = r (165) . We now evaluate the remaining finite z z z log N − r log T \log N-r\log T log N − r log T ( r , a ) (r,a) ( r , a )
λ r , T : = log N − r log T . (173) \lambda_{r,T}:=\log N-r\log T. \tag*{(173)} λ r , T := log N − r log T . (173) Since N = ⌊ T α ⌋ + 1 2 N=\lfloor T^\alpha\rfloor+\frac{1}{2} N = ⌊ T α ⌋ + 2 1
λ r , T = ( α − r ) log T + o ( log T ) . (174) \lambda_{r,T}=(\alpha-r)\log T+o(\log T). \tag*{(174)} λ r , T = ( α − r ) log T + o ( log T ) . (174) As α ∉ Z \alpha\notin\mathbb{Z} α ∈ / Z λ r , T \lambda_{r,T} λ r , T α − r \alpha-r α − r T T T ( r , a ) (r,a) ( r , a ) z z z
1 2 π i ∫ 2 σ 0 − 2 i T ′ 2 σ 0 + 2 i T ′ ( log T ) k − a e λ r , T z z − a − k − 1 J ~ r ( a ) ( z ; T ) d z . (175) \frac{1}{2\pi i}\int_{2\sigma_0-2iT'}^{2\sigma_0+2iT'}(\log T)^{k-a}e^{\lambda_{r,T}z}z^{-a-k-1}\widetilde{J}^{(a)}_r(z;T)\,dz. \tag*{(175)} 2 π i 1 ∫ 2 σ 0 − 2 i T ′ 2 σ 0 + 2 i T ′ ( log T ) k − a e λ r , T z z − a − k − 1 J r ( a ) ( z ; T ) d z . (175) As stated in (164) , J ~ r ( a ) ( z ; T ) \widetilde{J}^{(a)}_r(z;T) J r ( a ) ( z ; T ) z log T z\log T z log T z z z z = 0 z=0 z = 0
We first consider the case r > α r>\alpha r > α λ r , T < 0 \lambda_{r,T}<0 λ r , T < 0 T T T
ℜ z = 2 σ 0 , ∣ ℑ z ∣ ≤ 2 T ′ \Re z=2\sigma_0,\qquad|\Im z|\le2T' ℜ z = 2 σ 0 , ∣ℑ z ∣ ≤ 2 T ′ to the segment
ℜ z = A , ∣ ℑ z ∣ ≤ 2 T ′ , \Re z=A,\qquad|\Im z|\le2T', ℜ z = A , ∣ℑ z ∣ ≤ 2 T ′ , where A > 1 A>1 A > 1
∣ e λ r , T z ∣ = e A λ r , T ≪ T − A ( r − α ) + o ( 1 ) . (176) |e^{\lambda_{r,T}z}|=e^{A\lambda_{r,T}}\ll T^{-A(r-\alpha)+o(1)}. \tag*{(176)} ∣ e λ r , T z ∣ = e A λ r , T ≪ T − A ( r − α ) + o ( 1 ) . (176) Using (169) , we have
( log T ) k − a ∣ z ∣ − a − k − 1 ∣ J ~ r ( a ) ( z ; T ) ∣ ≪ k { ( log T ) k ∣ z ∣ − k − 1 , r = 0 , ( log T ) 2 k − 2 r ∣ z ∣ − 2 r − 1 , 1 ≤ r ≤ k . (177) (\log T)^{k-a}|z|^{-a-k-1}\left|\widetilde{J}^{(a)}_r(z;T)\right| \ll_k \begin{cases}(\log T)^k|z|^{-k-1}, & r=0,\\(\log T)^{2k-2r}|z|^{-2r-1}, & 1\le r\le k.\end{cases} \tag*{(177)} ( log T ) k − a ∣ z ∣ − a − k − 1 J r ( a ) ( z ; T ) ≪ k { ( log T ) k ∣ z ∣ − k − 1 , ( log T ) 2 k − 2 r ∣ z ∣ − 2 r − 1 , r = 0 , 1 ≤ r ≤ k . (177) On the line ℜ z = A \Re z=A ℜ z = A z z z (177) is absolutely integrable in ℑ z \Im z ℑ z ∣ ℑ z ∣ ≤ 2 T ′ |\Im z|\le2T' ∣ℑ z ∣ ≤ 2 T ′ log T \log T log T T − A ( r − α ) + o ( 1 ) T^{-A(r-\alpha)+o(1)} T − A ( r − α ) + o ( 1 ) o ( ( log T ) 2 k ) o((\log T)^{2k}) o (( log T ) 2 k ) ∣ ℑ z ∣ = 2 T ′ |\Im z|=2T' ∣ℑ z ∣ = 2 T ′ A A A (177) , their combined contribution is
≪ A { ( log T ) k ( T ′ ) − k − 1 , r = 0 , ( log T ) 2 k − 2 r ( T ′ ) − 2 r − 1 , 1 ≤ r ≤ k , = o ( ( log T ) 2 k ) . (178) \ll_A\begin{cases}(\log T)^k(T')^{-k-1}, & r=0,\\(\log T)^{2k-2r}(T')^{-2r-1}, & 1\le r\le k,\end{cases}=o((\log T)^{2k}). \tag*{(178)} ≪ A { ( log T ) k ( T ′ ) − k − 1 , ( log T ) 2 k − 2 r ( T ′ ) − 2 r − 1 , r = 0 , 1 ≤ r ≤ k , = o (( log T ) 2 k ) . (178) Therefore the whole ( r , a ) (r,a) ( r , a ) o ( ( log T ) 2 k ) o((\log T)^{2k}) o (( log T ) 2 k ) r > α r>\alpha r > α
We now consider the case r < α r<\alpha r < α λ r , T > 0 \lambda_{r,T}>0 λ r , T > 0
ℜ z = 2 σ 0 , ∣ ℑ z ∣ ≤ 2 T ′ , \Re z=2\sigma_0,\qquad|\Im z|\le2T', ℜ z = 2 σ 0 , ∣ℑ z ∣ ≤ 2 T ′ , to the segment
ℜ z = − A , ∣ ℑ z ∣ ≤ 2 T ′ , \Re z=-A,\qquad|\Im z|\le2T', ℜ z = − A , ∣ℑ z ∣ ≤ 2 T ′ , with the same fixed A > 1 A>1 A > 1 z = 0 z=0 z = 0
∣ e λ r , T z ∣ = e − A λ r , T ≪ T − A ( α − r ) + o ( 1 ) . (179) \left|e^{\lambda_{r,T}z}\right|=e^{-A\lambda_{r,T}}\ll T^{-A(\alpha-r)+o(1)}. \tag*{(179)} e λ r , T z = e − A λ r , T ≪ T − A ( α − r ) + o ( 1 ) . (179) Together with (177) , this shows that the new vertical side is o ( ( log T ) 2 k ) o((\log T)^{2k}) o (( log T ) 2 k ) o ( ( log T ) 2 k ) o((\log T)^{2k}) o (( log T ) 2 k )
1 2 π i ∫ 2 σ 0 − 2 i T ′ 2 σ 0 + 2 i T ′ ( log T ) k − a ( N T r ) z z − a − k − 1 J ~ r ( a ) ( z ; T ) d z = 1 r < α ( log T ) k − a Res z = 0 [ ( N T r ) z z − a − k − 1 J ~ r ( a ) ( z ; T ) ] + o ( ( log T ) 2 k ) . (180) \begin{aligned}
\frac{1}{2\pi i}\int_{2\sigma_0-2iT'}^{2\sigma_0+2iT'}(\log T)^{k-a}\left(\frac{N}{T^r}\right)^z z^{-a-k-1}\widetilde{J}^{(a)}_r(z;T)\,dz \\
&=\mathbf{1}_{r<\alpha}(\log T)^{k-a}\mathop{\operatorname{Res}}_{z=0}\left[\left(\frac{N}{T^r}\right)^z z^{-a-k-1}\widetilde{J}^{(a)}_r(z;T)\right]+o((\log T)^{2k}). \tag*{(180)}
\end{aligned} 2 π i 1 ∫ 2 σ 0 − 2 i T ′ 2 σ 0 + 2 i T ′ ( log T ) k − a ( T r N ) z z − a − k − 1 J r ( a ) ( z ; T ) d z = 1 r < α ( log T ) k − a Res z = 0 [ ( T r N ) z z − a − k − 1 J r ( a ) ( z ; T ) ] + o (( log T ) 2 k ) . (180) In order to evaluate the z z z z z z J ~ r ( a ) ( z ; T ) \widetilde{J}^{(a)}_r(z;T) J r ( a ) ( z ; T ) ( N / T r ) z (N/T^r)^z ( N / T r ) z
Res z = 0 [ exp ( z ( log N − r log T ) − z log T { ∑ j = 1 r ( x j − 1 ) + ∑ j = r + 1 k x j } ) z − a − k − 1 ] . (181) \mathop{\operatorname{Res}}_{z=0}\left[\exp\left(z(\log N-r\log T)-z\log T\left\{\sum_{j=1}^{r}(x_j-1)+\sum_{j=r+1}^{k}x_j\right\}\right)z^{-a-k-1}\right]. \tag*{(181)} Res z = 0 [ exp ( z ( log N − r log T ) − z log T { j = 1 ∑ r ( x j − 1 ) + j = r + 1 ∑ k x j } ) z − a − k − 1 ] . (181) Due to (161) , the above expression simplifies and we can evaluate the residue explicitly
Res z = 0 [ exp ( z ( log N − ( log T ) ∑ j = 1 k x j ) ) z − a − k − 1 ] = 1 ( a + k ) ! ( log N − ( log T ) ∑ j = 1 k x j ) a + k , \mathop{\operatorname{Res}}_{z=0}\left[\exp\left(z\left(\log N-(\log T)\sum_{j=1}^{k}x_j\right)\right)z^{-a-k-1}\right]=\frac{1}{(a+k)!}\left(\log N-(\log T)\sum_{j=1}^{k}x_j\right)^{a+k}, Res z = 0 [ exp ( z ( log N − ( log T ) j = 1 ∑ k x j ) ) z − a − k − 1 ] = ( a + k )! 1 ( log N − ( log T ) j = 1 ∑ k x j ) a + k , which gives
( log T ) k − a Res z = 0 [ ( N T r ) z z − a − k − 1 J ~ r ( a ) ( z ; T ) ] = ( log T ) k − a ( a + k ) ! ( 2 π i ) 2 k ∫ c j ( r ) ∫ c ^ ℓ ( r ) ( log N − ( log T ) ∑ j = 1 k x j ) a + k Δ ( x ) 2 Δ ( y ) 2 ∏ j , ℓ = 1 k ( x j − y ℓ − 1 ) (182) \begin{aligned}
(\log T)^{k-a}\mathop{\operatorname{Res}}_{z=0}\left[\left(\frac{N}{T^r}\right)^z z^{-a-k-1}\widetilde{J}^{(a)}_r(z;T)\right] \\
&=\frac{(\log T)^{k-a}}{(a+k)!(2\pi i)^{2k}}\int_{c_j(r)}\int_{\widehat{c}_\ell(r)}\left(\log N-(\log T)\sum_{j=1}^{k}x_j\right)^{a+k}\Delta(x)^2\Delta(y)^2\prod_{j,\ell=1}^{k}(x_j-y_\ell-1) \tag*{(182)}
\end{aligned} ( log T ) k − a Res z = 0 [ ( T r N ) z z − a − k − 1 J r ( a ) ( z ; T ) ] = ( a + k )! ( 2 π i ) 2 k ( log T ) k − a ∫ c j ( r ) ∫ c ℓ ( r ) ( log N − ( log T ) j = 1 ∑ k x j ) a + k Δ ( x ) 2 Δ ( y ) 2 j , ℓ = 1 ∏ k ( x j − y ℓ − 1 ) (182) × ( ∑ j = 1 k 1 x j + ∑ ℓ = 1 k 1 y ℓ + 1 ) a ( ∑ j = 1 k 1 x j − 1 + ∑ ℓ = 1 k 1 y ℓ ) k ∏ j = 1 k x j − k d x j ∏ ℓ = 1 k y ℓ − k d y ℓ . (183) \qquad\times\left(\sum_{j=1}^{k}\frac{1}{x_j}+\sum_{\ell=1}^{k}\frac{1}{y_\ell+1}\right)^a\left(\sum_{j=1}^{k}\frac{1}{x_j-1}+\sum_{\ell=1}^{k}\frac{1}{y_\ell}\right)^k\prod_{j=1}^{k}x_j^{-k}\,dx_j\prod_{\ell=1}^{k}y_\ell^{-k}\,dy_\ell. \tag*{(183)} × ( j = 1 ∑ k x j 1 + ℓ = 1 ∑ k y ℓ + 1 1 ) a ( j = 1 ∑ k x j − 1 1 + ℓ = 1 ∑ k y ℓ 1 ) k j = 1 ∏ k x j − k d x j ℓ = 1 ∏ k y ℓ − k d y ℓ . (183) As log N log T = α + o ( 1 ) \frac{\log N}{\log T}=\alpha+o(1) l o g T l o g N = α + o ( 1 ) x x x
( log T ) k − a ( log N − ( log T ) ∑ j = 1 k x j ) a + k = ( log T ) 2 k ( α − ∑ j = 1 k x j ) a + k + o ( ( log T ) 2 k ) (184) (\log T)^{k-a}\left(\log N-(\log T)\sum_{j=1}^{k}x_j\right)^{a+k}=(\log T)^{2k}\left(\alpha-\sum_{j=1}^{k}x_j\right)^{a+k}+o\left((\log T)^{2k}\right) \tag*{(184)} ( log T ) k − a ( log N − ( log T ) j = 1 ∑ k x j ) a + k = ( log T ) 2 k ( α − j = 1 ∑ k x j ) a + k + o ( ( log T ) 2 k ) (184) uniformly for x x x (180) –(184) with the preceding estimates gives
I k ( T , N ) = ( log T ) 2 k M k ( α ) + o ( ( log T ) 2 k ) , (185) I_k(T,N)=(\log T)^{2k}\mathcal{M}_k(\alpha)+o\left((\log T)^{2k}\right), \tag*{(185)} I k ( T , N ) = ( log T ) 2 k M k ( α ) + o ( ( log T ) 2 k ) , (185) where
M k ( α ) = 1 k ! 2 ∑ 0 ≤ r < α ( k r ) 2 ∑ a = r k ( k a ) 1 ( a + k ) ! × 1 ( 2 π i ) 2 k ∫ c j ( r ) ∫ c ^ ℓ ( r ) ( α − ∑ j = 1 k x j ) a + k Δ ( x ) 2 Δ ( y ) 2 ∏ j , ℓ = 1 k ( x j − y ℓ − 1 ) × ( ∑ j = 1 k 1 x j + ∑ ℓ = 1 k 1 y ℓ + 1 ) a ( ∑ j = 1 k 1 x j − 1 + ∑ ℓ = 1 k 1 y ℓ ) k ∏ j = 1 k x j − k d x j ∏ ℓ = 1 k y ℓ − k d y ℓ . \begin{aligned}
\mathcal{M}_k(\alpha)={}&\frac{1}{k!^2}\sum_{0\le r<\alpha}\binom{k}{r}^{2}\sum_{a=r}^{k}\binom{k}{a}\frac{1}{(a+k)!}\\
&\times\frac{1}{(2\pi i)^{2k}}\int_{c_j(r)}\int_{\widehat{c}_\ell(r)}
\left(\alpha-\sum_{j=1}^{k}x_j\right)^{a+k}\Delta(x)^2\Delta(y)^2\prod_{j,\ell=1}^{k}(x_j-y_\ell-1)\\
&\times\left(\sum_{j=1}^{k}\frac{1}{x_j}+\sum_{\ell=1}^{k}\frac{1}{y_\ell+1}\right)^a
\left(\sum_{j=1}^{k}\frac{1}{x_j-1}+\sum_{\ell=1}^{k}\frac{1}{y_\ell}\right)^k
\prod_{j=1}^{k}x_j^{-k}\,dx_j\prod_{\ell=1}^{k}y_\ell^{-k}\,dy_\ell.
\end{aligned} M k ( α ) = k ! 2 1 0 ≤ r < α ∑ ( r k ) 2 a = r ∑ k ( a k ) ( a + k )! 1 × ( 2 π i ) 2 k 1 ∫ c j ( r ) ∫ c ℓ ( r ) ( α − j = 1 ∑ k x j ) a + k Δ ( x ) 2 Δ ( y ) 2 j , ℓ = 1 ∏ k ( x j − y ℓ − 1 ) × ( j = 1 ∑ k x j 1 + ℓ = 1 ∑ k y ℓ + 1 1 ) a ( j = 1 ∑ k x j − 1 1 + ℓ = 1 ∑ k y ℓ 1 ) k j = 1 ∏ k x j − k d x j ℓ = 1 ∏ k y ℓ − k d y ℓ . Here the condition r < α r<\alpha r < α (180) ; no term with r > α r>\alpha r > α α ∉ Z \alpha\notin\mathbb{Z} α ∈ / Z r = α r=\alpha r = α N = ⌊ T α ⌋ + 1 2 N=\left\lfloor T^\alpha\right\rfloor+\frac{1}{2} N = ⌊ T α ⌋ + 2 1 T α T^\alpha T α O ( 1 ) O(1) O ( 1 ) T α T^\alpha T α ( − 1 ) k + a (-1)^{k+a} ( − 1 ) k + a ( α − ∑ j x j ) a + k \left(\alpha-\sum_j x_j\right)^{a+k} ( α − ∑ j x j ) a + k (39) .
Integer values for $\alpha. It remains to discuss the case in which α \alpha α α = m ∈ Z > 0 \alpha=m\in\mathbb{Z}_{>0} α = m ∈ Z > 0 r ≠ m r\ne m r = m
log N − r log T = ( m − r ) log T + o ( log T ) . \log N-r\log T=(m-r)\log T+o(\log T). log N − r log T = ( m − r ) log T + o ( log T ) . Thus the terms with r < m r<m r < m z = 0 z=0 z = 0 r > m r>m r > m r = m r=m r = m m ≤ a ≤ k m\le a\le k m ≤ a ≤ k m > k m>k m > k 1 ≤ m ≤ k 1\le m\le k 1 ≤ m ≤ k N = ⌊ T m ⌋ + 1 2 = T m + O ( 1 ) N=\left\lfloor T^m\right\rfloor+\frac{1}{2}=T^m+O(1) N = ⌊ T m ⌋ + 2 1 = T m + O ( 1 )
log N = log ( T m + O ( 1 ) ) = m log T + log ( 1 + O ( T − m ) ) , \log N=\log\left(T^m+O(1)\right)=m\log T+\log\left(1+O\left(T^{-m}\right)\right), log N = log ( T m + O ( 1 ) ) = m log T + log ( 1 + O ( T − m ) ) , which gives
log N − m log T = O ( T − m ) , (186) \log N-m\log T=O\left(T^{-m}\right), \tag*{(186)} log N − m log T = O ( T − m ) , (186) using log ( 1 + ϵ ) = O ( ϵ ) \log(1+\epsilon)=O(\epsilon) log ( 1 + ϵ ) = O ( ϵ ) ( N / T m ) z \left(N/T^m\right)^z ( N / T m ) z (186) is not of order log T \log T log T ( log T ) 2 k (\log T)^{2k} ( log T ) 2 k K T ′ ( z ) K_T'(z) K T ′ ( z ) 1 / z 1/z 1/ z ( m , a ) (m,a) ( m , a )
1 2 π i ∫ 2 σ 0 − 2 i T ′ 2 σ 0 + 2 i T ′ ( log T ) k − a ( N T m ) z z − a − k − 1 J ~ m ( a ) ( z ; T ) d z . \frac{1}{2\pi i}\int_{2\sigma_0-2iT'}^{2\sigma_0+2iT'}(\log T)^{k-a}\left(\frac{N}{T^m}\right)^z z^{-a-k-1}\widetilde{J}^{(a)}_m(z;T)\,dz. 2 π i 1 ∫ 2 σ 0 − 2 i T ′ 2 σ 0 + 2 i T ′ ( log T ) k − a ( T m N ) z z − a − k − 1 J m ( a ) ( z ; T ) d z . By (177) ,
( log T ) k − a ∣ z ∣ − a − k − 1 ∣ J ~ m ( a ) ( z ; T ) ∣ ≪ k ( log T ) 2 k − 2 m ∣ z ∣ − 2 m − 1 . (\log T)^{k-a}|z|^{-a-k-1}\left|\widetilde{J}^{(a)}_m(z;T)\right|\ll_k(\log T)^{2k-2m}|z|^{-2m-1}. ( log T ) k − a ∣ z ∣ − a − k − 1 J m ( a ) ( z ; T ) ≪ k ( log T ) 2 k − 2 m ∣ z ∣ − 2 m − 1 . Choose A T = T θ A_T=T^\theta A T = T θ 0 < θ < m 0<\theta<m 0 < θ < m ℜ z = A T \Re z=A_T ℜ z = A T
∣ ( N T m ) z ∣ = exp { O ( T − m A T ) } = 1 + o ( 1 ) , \left|\left(\frac{N}{T^m}\right)^z\right|=\exp\left\{O\left(T^{-m}A_T\right)\right\}=1+o(1), ( T m N ) z = exp { O ( T − m A T ) } = 1 + o ( 1 ) , the new vertical side is
≪ ( log T ) 2 k − 2 m ∫ − 2 T ′ 2 T ′ ( A T + ∣ y ∣ ) − 2 m − 1 d y ≪ ( log T ) 2 k − 2 m A T − 2 m = o ( ( log T ) 2 k ) . \ll(\log T)^{2k-2m}\int_{-2T'}^{2T'}(A_T+|y|)^{-2m-1}\,dy\ll(\log T)^{2k-2m}A_T^{-2m}=o\left((\log T)^{2k}\right). ≪ ( log T ) 2 k − 2 m ∫ − 2 T ′ 2 T ′ ( A T + ∣ y ∣ ) − 2 m − 1 d y ≪ ( log T ) 2 k − 2 m A T − 2 m = o ( ( log T ) 2 k ) . The two horizontal sides are also negligible, since their total contribution is
≪ ( log T ) 2 k − 2 m A T ( T ′ ) − 2 m − 1 = o ( ( log T ) 2 k ) . \ll(\log T)^{2k-2m}A_T(T')^{-2m-1}=o((\log T)^{2k}). ≪ ( log T ) 2 k − 2 m A T ( T ′ ) − 2 m − 1 = o (( log T ) 2 k ) . Thus the transition block r = m r=m r = m o ( ( log T ) 2 k ) o((\log T)^{2k}) o (( log T ) 2 k ) M k ( α ) M_k(\alpha) M k ( α ) α = m \alpha=m α = m r < m r<m r < m P r , k ( α ) P_{r,k}(\alpha) P r , k ( α ) α = r \alpha=r α = r 2 r 2 2r^2 2 r 2 M k ( α ) M_k(\alpha) M k ( α ) 2 r 2 − 1 2r^2-1 2 r 2 − 1 α = r \alpha=r α = r r = 1 , … , k r=1,\ldots,k r = 1 , … , k
Proof of Proposition 2.9
Throughout, we assume the hypotheses of Conjecture 1 ; by its application, the normalised moments in (42) are equal to the conjectural main term carrying an error term
O k , c , ξ ( ( log T ) 2 k − 1 + ξ ( 1 + min ( ∣ z ∣ , 1 ) log T ) − θ k , c ) . O_{k,c,\xi}\left((\log T)^{2k-1+\xi}(1+\min(|z|,1)\log T)^{-\theta_{k,c}}\right). O k , c , ξ ( ( log T ) 2 k − 1 + ξ ( 1 + min ( ∣ z ∣ , 1 ) log T ) − θ k , c ) . with θ k , c \theta_{k,c} θ k , c (125) . Following the two-centre decomposition, together with (162) , the main term derived in the proof of Theorem 2.2 is a finite linear combination of terms of the form
( log T ) k − a z − a − k J r ( a ) ( − z ; T ) = ( log T ) k − a z − a − k T − r z J ~ r ( a ) ( z ; T ) . (\log T)^{k-a}z^{-a-k}J_r^{(a)}(-z;T)=(\log T)^{k-a}z^{-a-k}T^{-rz}\widetilde{J}_r^{(a)}(z;T). ( log T ) k − a z − a − k J r ( a ) ( − z ; T ) = ( log T ) k − a z − a − k T − r z J r ( a ) ( z ; T ) . where 0 ≤ a , r ≤ k 0\leq a,r\leq k 0 ≤ a , r ≤ k 0 ≤ a ≤ k 0\leq a\leq k 0 ≤ a ≤ k J r ( a ) J_r^{(a)} J r ( a ) J ~ r ( a ) \widetilde{J}_r^{(a)} J r ( a ) (156) and (163) respectively. Since ℜ z > 0 \Re z>0 ℜ z > 0 ∣ T − r z ∣ ≤ 1 |T^{-rz}|\leq1 ∣ T − r z ∣ ≤ 1 Lemma 2.8 and through the steps outlined in (165) –(169) , we showed
J ~ r ( a ) ( z ; T ) = 0 ( a < r ) , \widetilde{J}_r^{(a)}(z;T)=0 \qquad(a<r), J r ( a ) ( z ; T ) = 0 ( a < r ) , and, for a ≥ r a\geq r a ≥ r
J ~ r ( a ) ( z ; T ) = { O k ( ( 1 + ∣ z ∣ log T ) a ) , r = 0 , O k ( ( 1 + ∣ z ∣ log T ) k + a − 2 r ) , 1 ≤ r ≤ k . (187) \widetilde{J}_r^{(a)}(z;T)=
\begin{cases}
O_k\left((1+|z|\log T)^a\right), & r=0,\\
O_k\left((1+|z|\log T)^{k+a-2r}\right), & 1\leq r\leq k.
\end{cases} \tag*{(187)} J r ( a ) ( z ; T ) = { O k ( ( 1 + ∣ z ∣ log T ) a ) , O k ( ( 1 + ∣ z ∣ log T ) k + a − 2 r ) , r = 0 , 1 ≤ r ≤ k . (187) It suffices to consider a ≥ r a\geq r a ≥ r
1 + min ( ∣ z ∣ , 1 ) log T ≍ 1 ∣ z ∣ log T ≤ 1. (188) 1+\min(|z|,1)\log T\asymp1 \qquad|z|\log T\leq1. \tag*{(188)} 1 + min ( ∣ z ∣ , 1 ) log T ≍ 1 ∣ z ∣ log T ≤ 1. (188) and
1 + 1 ∣ z ∣ log T ≍ 1 ∣ z ∣ log T > 1. (189) 1+\frac{1}{|z|\log T}\asymp1 \qquad|z|\log T>1. \tag*{(189)} 1 + ∣ z ∣ log T 1 ≍ 1 ∣ z ∣ log T > 1. (189) Suppose 1 ≤ r ≤ k 1\leq r\leq k 1 ≤ r ≤ k
( log T ) k − a ∣ z ∣ − a − k ∣ J ~ r ( a ) ( z ; T ) ∣ ≪ k ( log T ) k − a ∣ z ∣ − a − k ( 1 + ∣ z ∣ log T ) k + a − 2 r . (190) (\log T)^{k-a}|z|^{-a-k}\left|\widetilde{J}_r^{(a)}(z;T)\right|\ll_k(\log T)^{k-a}|z|^{-a-k}(1+|z|\log T)^{k+a-2r}. \tag*{(190)} ( log T ) k − a ∣ z ∣ − a − k J r ( a ) ( z ; T ) ≪ k ( log T ) k − a ∣ z ∣ − a − k ( 1 + ∣ z ∣ log T ) k + a − 2 r . (190) If ∣ z ∣ log T ≤ 1 |z|\log T\leq1 ∣ z ∣ log T ≤ 1
≪ k ( log T ) 2 k ( 1 + 1 ∣ z ∣ log T ) 2 k , \ll_k(\log T)^{2k}\left(1+\frac{1}{|z|\log T}\right)^{2k}, ≪ k ( log T ) 2 k ( 1 + ∣ z ∣ log T 1 ) 2 k , which follows from rewriting ∣ z ∣ − a − k = ( log T ) a + k ( ∣ z ∣ log T ) − a − k |z|^{-a-k}=(\log T)^{a+k}(|z|\log T)^{-a-k} ∣ z ∣ − a − k = ( log T ) a + k ( ∣ z ∣ log T ) − a − k 1 + ∣ z ∣ log T ≍ 1 1+|z|\log T\asymp1 1 + ∣ z ∣ log T ≍ 1 (42) . In the case ∣ z ∣ log T > 1 |z|\log T>1 ∣ z ∣ log T > 1
≪ k ( log T ) 2 k ( ∣ z ∣ log T ) − 2 r , \ll_k(\log T)^{2k}(|z|\log T)^{-2r}, ≪ k ( log T ) 2 k ( ∣ z ∣ log T ) − 2 r , using 1 + ∣ z ∣ log T ≍ ∣ z ∣ log T 1+|z|\log T\asymp|z|\log T 1 + ∣ z ∣ log T ≍ ∣ z ∣ log T r ≥ 1 r\geq1 r ≥ 1
≪ k ( log T ) 2 k ( 1 + min ( ∣ z ∣ , 1 ) log T ) − 2 , \ll_k(\log T)^{2k}(1+\min(|z|,1)\log T)^{-2}, ≪ k ( log T ) 2 k ( 1 + min ( ∣ z ∣ , 1 ) log T ) − 2 , and hence is again bounded by the right-hand side of (42) .
It remains to handle r = 0 r=0 r = 0 (187) ,
( log T ) k − a ∣ z ∣ − a − k ∣ J ~ 0 ( a ) ( z ; T ) ∣ ≪ k ( log T ) k − a ∣ z ∣ − a − k ( 1 + ∣ z ∣ log T ) a . (\log T)^{k-a}|z|^{-a-k}\left|\widetilde{J}_0^{(a)}(z;T)\right|\ll_k(\log T)^{k-a}|z|^{-a-k}(1+|z|\log T)^a. ( log T ) k − a ∣ z ∣ − a − k J 0 ( a ) ( z ; T ) ≪ k ( log T ) k − a ∣ z ∣ − a − k ( 1 + ∣ z ∣ log T ) a . If ∣ z ∣ log T ≤ 1 |z|\log T \le1 ∣ z ∣ log T ≤ 1 (42) immediately. If ∣ z ∣ log T > 1 |z|\log T > 1 ∣ z ∣ log T > 1
≪ k ( log T ) 2 k ( ∣ z ∣ log T ) − k . \ll_k (\log T)^{2k}(|z|\log T)^{-k}. ≪ k ( log T ) 2 k ( ∣ z ∣ log T ) − k . For k ≥ 2 k \ge2 k ≥ 2
≪ k ( log T ) 2 k ( 1 + min ( ∣ z ∣ , 1 ) log T ) − 2 . \ll_k (\log T)^{2k}\left(1+\min(|z|,1)\log T\right)^{-2}. ≪ k ( log T ) 2 k ( 1 + min ( ∣ z ∣ , 1 ) log T ) − 2 . The only remaining case is when k = 1 k=1 k = 1 r = 0 r=0 r = 0 (163) gives
J ~ 0 ( 0 ) ( z ; T ) = 0 , J ~ 0 ( 1 ) ( z ; T ) = 1. \widetilde{J}_0^{(0)}(z;T)=0,\qquad\widetilde{J}_0^{(1)}(z;T)=1. J 0 ( 0 ) ( z ; T ) = 0 , J 0 ( 1 ) ( z ; T ) = 1. Thus the r = 0 r=0 r = 0 O ( ∣ z ∣ − 2 ) O(|z|^{-2}) O ( ∣ z ∣ − 2 ) k = 1 k=1 k = 1 ∣ z ∣ log T ≤ 1 |z|\log T \le1 ∣ z ∣ log T ≤ 1
∣ z ∣ − 2 ≪ ( log T ) 2 ( 1 + 1 ∣ z ∣ log T ) 2 , |z|^{-2}\ll(\log T)^2\left(1+\frac{1}{|z|\log T}\right)^2, ∣ z ∣ − 2 ≪ ( log T ) 2 ( 1 + ∣ z ∣ log T 1 ) 2 , while if ∣ z ∣ log T > 1 |z|\log T>1 ∣ z ∣ log T > 1
∣ z ∣ − 2 ≪ ( log T ) 2 ( 1 + min ( ∣ z ∣ , 1 ) log T ) − 2 . |z|^{-2}\ll(\log T)^2\left(1+\min(|z|,1)\log T\right)^{-2}. ∣ z ∣ − 2 ≪ ( log T ) 2 ( 1 + min ( ∣ z ∣ , 1 ) log T ) − 2 . Hence the k = 1 k=1 k = 1 r = 0 r=0 r = 0 (42) . Therefore, the conjectural main term satisfies (42) for every k ≥ 1 k\ge1 k ≥ 1 ∣ z ∣ log T ≤ 1 |z|\log T\le1 ∣ z ∣ log T ≤ 1 (188) holds and the error term is
≪ k , c , ξ ( log T ) 2 k − 1 + ξ . \ll_{k,c,\xi}(\log T)^{2k-1+\xi}. ≪ k , c , ξ ( log T ) 2 k − 1 + ξ . As ξ < 1 \xi<1 ξ < 1 o ( ( log T ) 2 k ) o((\log T)^{2k}) o (( log T ) 2 k ) (42) . Suppose now that ∣ z ∣ log T > 1 |z|\log T>1 ∣ z ∣ log T > 1
( log T ) 2 k ( 1 + min ( ∣ z ∣ , 1 ) log T ) − 2 ⋅ ( log T ) − 1 + ξ ( 1 + min ( ∣ z ∣ , 1 ) log T ) 2 − θ k , c . (\log T)^{2k}\left(1+\min(|z|,1)\log T\right)^{-2}\cdot(\log T)^{-1+\xi}\left(1+\min(|z|,1)\log T\right)^{2-\theta_{k,c}}. ( log T ) 2 k ( 1 + min ( ∣ z ∣ , 1 ) log T ) − 2 ⋅ ( log T ) − 1 + ξ ( 1 + min ( ∣ z ∣ , 1 ) log T ) 2 − θ k , c . If k ≥ 3 k\ge3 k ≥ 3 θ k , c = k − 1 ≥ 2 \theta_{k,c}=k-1\ge2 θ k , c = k − 1 ≥ 2 O ( 1 ) O(1) O ( 1 ) 0 < ξ < 1 0<\xi<1 0 < ξ < 1 k = 1 , 2 k=1,2 k = 1 , 2 θ k , c = 1 + c \theta_{k,c}=1+c θ k , c = 1 + c
≪ ( log T ) − 1 + ξ ( log T ) 1 − c = ( log T ) ξ − c ≪ 1 , \ll(\log T)^{-1+\xi}(\log T)^{1-c}=(\log T)^{\xi-c}\ll1, ≪ ( log T ) − 1 + ξ ( log T ) 1 − c = ( log T ) ξ − c ≪ 1 , because 0 < ξ < c 0<\xi<c 0 < ξ < c (42) . Finally, on any sub-range satisfying ∣ z ∣ log T ≥ c 0 > 0 |z|\log T\ge c_0>0 ∣ z ∣ log T ≥ c 0 > 0
( 1 + 1 ∣ z ∣ log T ) 2 k ≪ k , c 0 1 , \left(1+\frac{1}{|z|\log T}\right)^{2k}\ll_{k,c_0}1, ( 1 + ∣ z ∣ log T 1 ) 2 k ≪ k , c 0 1 , and (43) follows.
Proof of Proposition 2.10
To determine the zero-order of P r , k ( α ) P_{r,k}(\alpha) P r , k ( α ) r = α r=\alpha r = α x j − 1 x_j-1 x j − 1 1 ≤ j ≤ r 1\le j\le r 1 ≤ j ≤ r y ℓ + 1 y_\ell+1 y ℓ + 1 1 ≤ ℓ ≤ r 1\le\ell\le r 1 ≤ ℓ ≤ r x j x_j x j r < j ≤ k r<j\le k r < j ≤ k y ℓ y_\ell y ℓ r < ℓ ≤ k r<\ell\le k r < ℓ ≤ k
∑ j = 1 k x j − α = ∑ j = 1 r ( x j − 1 ) + ∑ j = r + 1 k x j − ( α − r ) , \sum_{j=1}^{k}x_j-\alpha=\sum_{j=1}^{r}(x_j-1)+\sum_{j=r+1}^{k}x_j-(\alpha-r), j = 1 ∑ k x j − α = j = 1 ∑ r ( x j − 1 ) + j = r + 1 ∑ k x j − ( α − r ) , we have
( ∑ j = 1 k x j − α ) k + m = ∑ d = 0 k + m ( k + m d ) ( ∑ j = 1 r ( x j − 1 ) + ∑ j = r + 1 k x j ) d ( − 1 ) k + m − d ( α − r ) k + m − d . \left(\sum_{j=1}^{k}x_j-\alpha\right)^{k+m}
=\sum_{d=0}^{k+m}\binom{k+m}{d}\left(\sum_{j=1}^{r}(x_j-1)+\sum_{j=r+1}^{k}x_j\right)^d(-1)^{k+m-d}(\alpha-r)^{k+m-d}. ( j = 1 ∑ k x j − α ) k + m = d = 0 ∑ k + m ( d k + m ) ( j = 1 ∑ r ( x j − 1 ) + j = r + 1 ∑ k x j ) d ( − 1 ) k + m − d ( α − r ) k + m − d . We claim that the residue corresponding to the degree-d d d d > k + m − 2 r 2 d>k+m-2r^2 d > k + m − 2 r 2
Δ ( x ) 2 Δ ( y ) 2 ∏ i , j = 1 k ( x i − y j − 1 ) ∏ j = 1 k x j − k ∏ ℓ = 1 k y ℓ − k , \Delta(x)^2\Delta(y)^2\prod_{i,j=1}^{k}(x_i-y_j-1)\prod_{j=1}^{k}x_j^{-k}\prod_{\ell=1}^{k}y_\ell^{-k}, Δ ( x ) 2 Δ ( y ) 2 i , j = 1 ∏ k ( x i − y j − 1 ) j = 1 ∏ k x j − k ℓ = 1 ∏ k y ℓ − k , from the integrand in (39) . In all 2 k 2k 2 k
2 r ( r − 1 ) + 2 ( k − r ) ( k − r − 1 ) + 2 r ( k − r ) − 2 k ( k − r ) = 2 r 2 − 2 k , 2r(r-1)+2(k-r)(k-r-1)+2r(k-r)-2k(k-r)=2r^2-2k, 2 r ( r − 1 ) + 2 ( k − r ) ( k − r − 1 ) + 2 r ( k − r ) − 2 k ( k − r ) = 2 r 2 − 2 k , where the contribution 2 r ( k − r ) 2r(k-r) 2 r ( k − r ) x i − y j − 1 x_i-y_j-1 x i − y j − 1 1 ≤ i ≤ r 1\le i\le r 1 ≤ i ≤ r r < j ≤ k r<j\le k r < j ≤ k r < i ≤ k r<i\le k r < i ≤ k 1 ≤ j ≤ r 1\le j\le r 1 ≤ j ≤ r
( ∑ j = 1 k 1 x j + ∑ ℓ = 1 k 1 y ℓ + 1 ) m ( ∑ j = 1 k 1 x j − 1 + ∑ ℓ = 1 k 1 y ℓ ) k \left(\sum_{j=1}^{k}\frac{1}{x_j}+\sum_{\ell=1}^{k}\frac{1}{y_\ell+1}\right)^m
\left(\sum_{j=1}^{k}\frac{1}{x_j-1}+\sum_{\ell=1}^{k}\frac{1}{y_\ell}\right)^k ( j = 1 ∑ k x j 1 + ℓ = 1 ∑ k y ℓ + 1 1 ) m ( j = 1 ∑ k x j − 1 1 + ℓ = 1 ∑ k y ℓ 1 ) k has total pole order at most m + k m+k m + k d d d
( ∑ j = 1 k x j − α ) k + m , \left(\sum_{j=1}^{k}x_j-\alpha\right)^{k+m}, ( j = 1 ∑ k x j − α ) k + m , the local total degree of the full integrand is at least
d + ( 2 r 2 − 2 k ) − m − k = d + 2 r 2 − 3 k − m . d+(2r^2-2k)-m-k=d+2r^2-3k-m. d + ( 2 r 2 − 2 k ) − m − k = d + 2 r 2 − 3 k − m . A non-zero residue results only from monomials of the form
∏ j = 1 r ( x j − 1 ) − 1 ∏ j = r + 1 k x j − 1 ∏ ℓ = 1 r ( y ℓ + 1 ) − 1 ∏ ℓ = r + 1 k y ℓ − 1 , \prod_{j=1}^{r}(x_j-1)^{-1}\prod_{j=r+1}^{k}x_j^{-1}
\prod_{\ell=1}^{r}(y_\ell+1)^{-1}\prod_{\ell=r+1}^{k}y_\ell^{-1}, j = 1 ∏ r ( x j − 1 ) − 1 j = r + 1 ∏ k x j − 1 ℓ = 1 ∏ r ( y ℓ + 1 ) − 1 ℓ = r + 1 ∏ k y ℓ − 1 , which has total degree − 2 k -2k − 2 k
d + 2 r 2 − 3 k − m > − 2 k , d+2r^2-3k-m>-2k, d + 2 r 2 − 3 k − m > − 2 k , or equivalently whenever d > k + m − 2 r 2 d>k+m-2r^2 d > k + m − 2 r 2 ( α − r ) q (\alpha-r)^q ( α − r ) q m m m (39) corresponds to d = k + m − q d=k+m-q d = k + m − q q < 2 r 2 q<2r^2 q < 2 r 2
d = k + m − q > k + m − 2 r 2 , d=k+m-q>k+m-2r^2, d = k + m − q > k + m − 2 r 2 , so that coefficient vanishes. Thus every coefficient of ( α − r ) q (\alpha-r)^q ( α − r ) q q < 2 r 2 q<2r^2 q < 2 r 2 P r , k ( α ) P_{r,k}(\alpha) P r , k ( α ) P r , k ( α ) P_{r,k}(\alpha) P r , k ( α ) ( α − r ) 2 r 2 (\alpha-r)^{2r^2} ( α − r ) 2 r 2
P r , k ( α ) = O ( ( α − r ) 2 r 2 ) ( α → r ) , P_{r,k}(\alpha)=O\left((\alpha-r)^{2r^2}\right)\qquad(\alpha\to r), P r , k ( α ) = O ( ( α − r ) 2 r 2 ) ( α → r ) , as required.
Proof of Theorem 2.3
Let integer k ≥ 1 k\ge1 k ≥ 1 H = X 1 − 1 / α H=X^{1-1/\alpha} H = X 1 − 1/ α α > 0 \alpha>0 α > 0 g ∈ C c ∞ ( ( 0 , ∞ ) ) g\in C^\infty_c((0,\infty)) g ∈ C c ∞ (( 0 , ∞ )) supp g ⊂ [ 1 / 2 , 4 ] \operatorname{supp}g\subset[1/2,4] supp g ⊂ [ 1/2 , 4 ]
V ~ g , k ( X , H ) : = 1 H X ∫ R g ( x X ) ∣ Δ k ( x ; H ) ∣ 2 d x . (191) \widetilde V_{g,k}(X,H):=\frac{1}{HX}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)\left|\Delta_k(x;H)\right|^2\,dx. \tag*{(191)} V g , k ( X , H ) := H X 1 ∫ R g ( X x ) ∣ Δ k ( x ; H ) ∣ 2 d x . (191) The support condition localises x x x x ≍ X x\asymp X x ≍ X x ∈ [ X , 2 X ] x\in[X,2X] x ∈ [ X , 2 X ] 1 [ 1 , 2 ] ( u ) \mathbf{1}_{[1,2]}(u) 1 [ 1 , 2 ] ( u ) [ 1 / 2 , 4 ] [1/2,4] [ 1/2 , 4 ]
lim X → ∞ V ~ g , k ( X , X 1 − 1 / α ) ( log ( X 1 / α ) ) 2 k − 1 = ∫ R g ( u ) d u 1 k ! 2 ∑ 0 ≤ r < α ( k r ) 2 P r , k ′ ( α ) , (192) \lim_{X\to\infty}\frac{\widetilde V_{g,k}(X,X^{1-1/\alpha})}{(\log(X^{1/\alpha}))^{2k-1}}
=\int_{\mathbb{R}}g(u)\,du\,\frac{1}{k!^2}\sum_{0\le r<\alpha}\binom{k}{r}^2P'_{r,k}(\alpha), \tag*{(192)} X → ∞ lim ( log ( X 1/ α ) ) 2 k − 1 V g , k ( X , X 1 − 1/ α ) = ∫ R g ( u ) d u k ! 2 1 0 ≤ r < α ∑ ( r k ) 2 P r , k ′ ( α ) , (192) for every fixed non-negative g ∈ C c ∞ ( ( 0 , ∞ ) ) g\in C^\infty_c((0,\infty)) g ∈ C c ∞ (( 0 , ∞ )) supp g ⊂ [ 1 / 2 , 4 ] \operatorname{supp}g\subset[1/2,4] supp g ⊂ [ 1/2 , 4 ] [32 ] ) to show that a truncated Perron integral on the line ℜ s = c \Re s=c ℜ s = c
c = 1 + 1 log X c=1+\frac{1}{\log X} c = 1 + log X 1 approximates Ψ k ( x ; H ) \Psi_k(x;H) Ψ k ( x ; H ) (17) with a negligible mean-square error. Note that this choice of c c c (8) while keeping X c − 1 = O ( 1 ) X^{c-1}=O(1) X c − 1 = O ( 1 ) − ζ ′ / ζ ( c ) ≪ log X -\zeta'/\zeta(c)\ll\log X − ζ ′ / ζ ( c ) ≪ log X (127) . This ensures that the truncation errors remain only logarithmic in size. Define the truncated Perron kernel
K T ( u ) : = 1 2 π i ∫ c − i T c + i T u s d s s . K_T(u) := \frac{1}{2\pi i}\int_{c-iT}^{c+iT}u^s\frac{ds}{s}. K T ( u ) := 2 π i 1 ∫ c − i T c + i T u s s d s . Then
1 2 π i ∫ c − i T c + i T ( − ζ ′ ζ ( s ) ) k ( x + H ) s − x s s d s = ∑ n = 1 ∞ Λ ∗ k ( n ) [ K T ( x + H n ) − K T ( x n ) ] . (193) \frac{1}{2\pi i}\int_{c-iT}^{c+iT}\left(-\frac{\zeta'}{\zeta}(s)\right)^k\frac{(x+H)^s-x^s}{s}\,ds=\sum_{n=1}^{\infty}\Lambda^{*k}(n)\left[K_T\left(\frac{x+H}{n}\right)-K_T\left(\frac{x}{n}\right)\right]. \tag*{(193)} 2 π i 1 ∫ c − i T c + i T ( − ζ ζ ′ ( s ) ) k s ( x + H ) s − x s d s = n = 1 ∑ ∞ Λ ∗ k ( n ) [ K T ( n x + H ) − K T ( n x ) ] . (193) By Lemma 3.2, equivalently by the usual truncated Perron kernel estimate,
K T ( u ) = 1 u > 1 + O ( u c min { 1 , 1 T ∣ log u ∣ } ) . (194) K_T(u)=1_{u>1}+O\left(u^c\min\left\{1,\frac{1}{T|\log u|}\right\}\right). \tag*{(194)} K T ( u ) = 1 u > 1 + O ( u c min { 1 , T ∣ log u ∣ 1 } ) . (194) This means
K T ( x + H n ) − K T ( x n ) K_T\left(\frac{x+H}{n}\right)-K_T\left(\frac{x}{n}\right) K T ( n x + H ) − K T ( n x ) approximates the indicator of x < n ≤ x + H x<n\le x+H x < n ≤ x + H x x x x + H x+H x + H (194) measures the error made by replacing the sharp cutoff 1 u > 1 1_{u>1} 1 u > 1 K T ( u ) K_T(u) K T ( u ) T T T V ~ g , k ( X , H ) \widetilde{V}_{g,k}(X,H) V g , k ( X , H )
T = X 1 + ϵ H , T=\frac{X^{1+\epsilon}}{\sqrt{H}}, T = H X 1 + ϵ , and choose ϵ > 0 \epsilon>0 ϵ > 0 H X HX H X o ( ( log ( X 1 / α ) ) 2 k − 1 ) o((\log(X^{1/\alpha}))^{2k-1}) o (( log ( X 1/ α ) ) 2 k − 1 ) (192) . More precisely, since H = X 1 − 1 / α H=X^{1-1/\alpha} H = X 1 − 1/ α X T H = o ( 1 ) \frac{X}{TH}=o(1) T H X = o ( 1 )
X 1 2 α − 1 2 − ϵ = o ( 1 ) . X^{\frac{1}{2\alpha}-\frac{1}{2}-\epsilon}=o(1). X 2 α 1 − 2 1 − ϵ = o ( 1 ) . Thus it is enough to assume ϵ > max { 0 , 1 2 α − 1 2 } \epsilon>\max\left\{0,\frac{1}{2\alpha}-\frac{1}{2}\right\} ϵ > max { 0 , 2 α 1 − 2 1 } (194) requires care is the range where one of the arguments of K T K_T K T u = 1 u=1 u = 1 g ( x / X ) g(x/X) g ( x / X ) x ∈ [ X / 2 , 4 X ] x\in[X/2,4X] x ∈ [ X /2 , 4 X ] x ≍ X x\asymp X x ≍ X H = o ( X ) H=o(X) H = o ( X ) ( x , x + H ] (x,x+H] ( x , x + H ] X X X n n n
x < n ≤ x + H , x<n\le x+H, x < n ≤ x + H , must also satisfy
A X ≤ n ≤ B X AX\le n\le BX A X ≤ n ≤ B X for suitable fixed constants A , B > 0 A,B>0 A , B > 0 supp g \operatorname{supp}g supp g n ∈ [ A X , B X ] n\in[AX,BX] n ∈ [ A X , B X ] (17) from the truncated Perron expression in (193) and splitting the resulting sum into the near and outer ranges gives
1 2 π i ∫ c − i T c + i T ( − ζ ′ ζ ( s ) ) k ( x + H ) s − x s s d s − Ψ k ( x ; H ) = E n e a r ( x ) + E o u t ( x ) , \frac{1}{2\pi i}\int_{c-iT}^{c+iT}\left(-\frac{\zeta'}{\zeta}(s)\right)^k\frac{(x+H)^s-x^s}{s}\,ds-\Psi_k(x;H)=\mathcal{E}_{\mathrm{near}}(x)+\mathcal{E}_{\mathrm{out}}(x), 2 π i 1 ∫ c − i T c + i T ( − ζ ζ ′ ( s ) ) k s ( x + H ) s − x s d s − Ψ k ( x ; H ) = E near ( x ) + E out ( x ) , where
E n e a r ( x ) : = ∑ n ∈ [ A X , B X ] Λ ∗ k ( n ) [ K T ( x + H n ) − K T ( x n ) − 1 { x < n ≤ x + H } ] . (195) \mathcal{E}_{\mathrm{near}}(x):=\sum_{n\in[AX,BX]}\Lambda^{*k}(n)\left[K_T\left(\frac{x+H}{n}\right)-K_T\left(\frac{x}{n}\right)-1_{\{x<n\le x+H\}}\right]. \tag*{(195)} E near ( x ) := n ∈ [ A X , B X ] ∑ Λ ∗ k ( n ) [ K T ( n x + H ) − K T ( n x ) − 1 { x < n ≤ x + H } ] . (195) and
E o u t ( x ) : = ∑ n ∉ [ A X , B X ] Λ ∗ k ( n ) [ K T ( x + H n ) − K T ( x n ) ] . (196) \mathcal{E}_{\mathrm{out}}(x):=\sum_{n\notin[AX,BX]}\Lambda^{*k}(n)\left[K_T\left(\frac{x+H}{n}\right)-K_T\left(\frac{x}{n}\right)\right]. \tag*{(196)} E out ( x ) := n ∈ / [ A X , B X ] ∑ Λ ∗ k ( n ) [ K T ( n x + H ) − K T ( n x ) ] . (196) In the outer range, the indicator 1 x < n ≤ x + H \mathbf{1}_{x<n\leq x+H} 1 x < n ≤ x + H x / n x/n x / n ( x + H ) / n (x+H)/n ( x + H ) / n
∣ K T ( x + H n ) − K T ( x n ) ∣ ≪ 1 T [ ( x + H n ) c + ( x n ) c ] ≪ X c T n c , \left|K_T\left(\frac{x+H}{n}\right)-K_T\left(\frac{x}{n}\right)\right|\ll\frac{1}{T}\left[\left(\frac{x+H}{n}\right)^c+\left(\frac{x}{n}\right)^c\right]\ll\frac{X^c}{Tn^c}, K T ( n x + H ) − K T ( n x ) ≪ T 1 [ ( n x + H ) c + ( n x ) c ] ≪ T n c X c , using (194) for the outer range. By the triangle inequality and the preceding estimate,
∣ E o u t ( x ) ∣ ≤ ∑ n ∉ [ A X , B X ] Λ ∗ k ( n ) ∣ K T ( x + H n ) − K T ( x n ) ∣ ≪ X c T ∑ n ∉ [ A X , B X ] Λ ∗ k ( n ) n c ≤ X c T ∑ n = 1 ∞ Λ ∗ k ( n ) n c . \begin{aligned}
|\mathcal{E}_{\mathrm{out}}(x)|&\leq\sum_{n\notin[AX,BX]}\Lambda^{*k}(n)\left|K_T\left(\frac{x+H}{n}\right)-K_T\left(\frac{x}{n}\right)\right|\\
&\ll\frac{X^c}{T}\sum_{n\notin[AX,BX]}\frac{\Lambda^{*k}(n)}{n^c}\\
&\leq\frac{X^c}{T}\sum_{n=1}^{\infty}\frac{\Lambda^{*k}(n)}{n^c}.
\end{aligned} ∣ E out ( x ) ∣ ≤ n ∈ / [ A X , B X ] ∑ Λ ∗ k ( n ) K T ( n x + H ) − K T ( n x ) ≪ T X c n ∈ / [ A X , B X ] ∑ n c Λ ∗ k ( n ) ≤ T X c n = 1 ∑ ∞ n c Λ ∗ k ( n ) . Since Λ ∗ k ( n ) ≥ 0 \Lambda^{*k}(n)\geq0 Λ ∗ k ( n ) ≥ 0 c = 1 + 1 / log X c=1+1/\log X c = 1 + 1/ log X
∑ n = 1 ∞ Λ ∗ k ( n ) n c = ( ∑ n = 1 ∞ Λ ( n ) n c ) k = ( − ζ ′ ζ ( c ) ) k ≪ k ( log X ) k , \sum_{n=1}^{\infty}\frac{\Lambda^{*k}(n)}{n^c}=\left(\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n^c}\right)^k=\left(-\frac{\zeta'}{\zeta}(c)\right)^k\ll_k(\log X)^k, n = 1 ∑ ∞ n c Λ ∗ k ( n ) = ( n = 1 ∑ ∞ n c Λ ( n ) ) k = ( − ζ ζ ′ ( c ) ) k ≪ k ( log X ) k , as in (128) . The approximation in (193) will be inserted into the weighted mean square (191) . For δ : = X / T \delta:=X/T δ := X / T (191) are bounded by
E o u t ( x ) ≪ k δ ( log X ) k , ∫ R g ( x X ) ∣ E o u t ( x ) ∣ 2 d x ≪ k X δ 2 ( log X ) 2 k . \mathcal{E}_{\mathrm{out}}(x)\ll_k\delta(\log X)^k,\qquad\int_{\mathbb{R}}g\left(\frac{x}{X}\right)|\mathcal{E}_{\mathrm{out}}(x)|^2\,dx\ll_k X\delta^2(\log X)^{2k}. E out ( x ) ≪ k δ ( log X ) k , ∫ R g ( X x ) ∣ E out ( x ) ∣ 2 d x ≪ k X δ 2 ( log X ) 2 k . It remains to handle the near range n ≍ X n\asymp X n ≍ X
∣ K T ( x + H n ) − K T ( x n ) − 1 x < n ≤ x + H ∣ ≪ min { 1 , δ ∣ x − n ∣ } + min { 1 , δ ∣ x + H − n ∣ } . (197) \left|K_T\left(\frac{x+H}{n}\right)-K_T\left(\frac{x}{n}\right)-\mathbf{1}_{x<n\leq x+H}\right|\ll\min\left\{1,\frac{\delta}{|x-n|}\right\}+\min\left\{1,\frac{\delta}{|x+H-n|}\right\}. \tag*{(197)} K T ( n x + H ) − K T ( n x ) − 1 x < n ≤ x + H ≪ min { 1 , ∣ x − n ∣ δ } + min { 1 , ∣ x + H − n ∣ δ } . (197) The near-range error arises from the truncated Perron kernel being least accurate near the jump points of the approximations for the two indicators 1 n ≤ x \mathbf{1}_{n\leq x} 1 n ≤ x 1 n ≤ x + H \mathbf{1}_{n\leq x+H} 1 n ≤ x + H x = n x=n x = n x = n − H x=n-H x = n − H
K δ ( y ) : = { min { 1 , δ ∣ y ∣ } , y ≠ 0 , 1 , y = 0. \mathcal{K}_{\delta}(y):=\begin{cases}\min\left\{1,\dfrac{\delta}{|y|}\right\},&y\ne0,\\1,&y=0.\end{cases} K δ ( y ) := ⎩ ⎨ ⎧ min { 1 , ∣ y ∣ δ } , 1 , y = 0 , y = 0. Using (197) , E n e a r ( x ) \mathcal{E}_{\mathrm{near}}(x) E near ( x ) (195) and the triangle inequality, we have
∣ E n e a r ( x ) ∣ ≪ ∑ n ≍ X Λ ∗ k ( n ) K δ ( x − n ) + ∑ n ≍ X Λ ∗ k ( n ) K δ ( x + H − n ) . |\mathcal{E}_{\mathrm{near}}(x)|\ll\sum_{n\asymp X}\Lambda^{*k}(n)\mathcal{K}_{\delta}(x-n)+\sum_{n\asymp X}\Lambda^{*k}(n)\mathcal{K}_{\delta}(x+H-n). ∣ E near ( x ) ∣ ≪ n ≍ X ∑ Λ ∗ k ( n ) K δ ( x − n ) + n ≍ X ∑ Λ ∗ k ( n ) K δ ( x + H − n ) . These are the two boundary contributions, centred respectively at x = n x=n x = n x = n − H x=n-H x = n − H x x x H H H n ≍ X n\asymp X n ≍ X x ≍ X x\asymp X x ≍ X
∫ x ≍ X K δ ( x − n ) d x ≪ δ log X , ∑ n ≍ X K δ ( x − n ) ≪ ( 1 + δ ) log X . \int_{x\asymp X}\mathcal{K}_{\delta}(x-n)\,dx\ll\delta\log X,\qquad\sum_{n\asymp X}\mathcal{K}_{\delta}(x-n)\ll(1+\delta)\log X. ∫ x ≍ X K δ ( x − n ) d x ≪ δ log X , n ≍ X ∑ K δ ( x − n ) ≪ ( 1 + δ ) log X . By (126) , together with the standard estimate ∑ n ≤ Y d k ( n ) 2 ≪ k Y ( log Y ) k 2 − 1 \sum_{n\leq Y}d_k(n)^2\ll_kY(\log Y)^{k^2-1} ∑ n ≤ Y d k ( n ) 2 ≪ k Y ( log Y ) k 2 − 1 [21 §1.6, (1.80)] , we obtain
∑ n ≤ Y ∣ Λ ∗ k ( n ) ∣ 2 ≪ k Y ( log Y ) B k (198) \sum_{n\leq Y}|\Lambda^{*k}(n)|^2\ll_kY(\log Y)^{B_k} \tag*{(198)} n ≤ Y ∑ ∣ Λ ∗ k ( n ) ∣ 2 ≪ k Y ( log Y ) B k (198) for some constant B k > 0 B_k > 0 B k > 0 [17 Theorem 5.2, p. 22] , applied with counting measure in n n n x x x
∫ R ∣ g ( x X ) ∣ ∣ ∑ n ≍ X Λ ∗ k ( n ) K δ ( x − n ) ∣ 2 d x \int_{\mathbb{R}} \left|g\left(\frac{x}{X}\right)\right| \left|\sum_{n\asymp X} \Lambda^{*k}(n)\mathcal{K}_{\delta}(x-n)\right|^2 dx ∫ R g ( X x ) n ≍ X ∑ Λ ∗ k ( n ) K δ ( x − n ) 2 d x ≪ g ( sup x ≍ X ∑ n ≍ X K δ ( x − n ) ) ( sup n ≍ X ∫ x ≍ X K δ ( x − n ) d x ) ∑ n ≍ X ∣ Λ ∗ k ( n ) ∣ 2 ≪ ( 1 + δ ) log X ⋅ δ log X ∑ n ≍ X ∣ Λ ∗ k ( n ) ∣ 2 \begin{aligned}
\ll_g \left(\sup_{x\asymp X}\sum_{n\asymp X}\mathcal{K}_{\delta}(x-n)\right)\left(\sup_{n\asymp X}\int_{x\asymp X}\mathcal{K}_{\delta}(x-n)\,dx\right)\sum_{n\asymp X}|\Lambda^{*k}(n)|^2 \\
&\ll(1+\delta)\log X\cdot\delta\log X\sum_{n\asymp X}|\Lambda^{*k}(n)|^2
\end{aligned} ≪ g ( x ≍ X sup n ≍ X ∑ K δ ( x − n ) ) ( n ≍ X sup ∫ x ≍ X K δ ( x − n ) d x ) n ≍ X ∑ ∣ Λ ∗ k ( n ) ∣ 2 ≪ ( 1 + δ ) log X ⋅ δ log X n ≍ X ∑ ∣ Λ ∗ k ( n ) ∣ 2 ≪ k X ( δ + δ 2 ) ( log X ) B k , \ll_k X(\delta+\delta^2)(\log X)^{B_k}, ≪ k X ( δ + δ 2 ) ( log X ) B k , where in the last line we applied (198) and also enlarged B k B_k B k
∫ R ∣ g ( x X ) ∣ ∣ ∑ n ≍ X Λ ∗ k ( n ) K δ ( x + H − n ) ∣ 2 d x ≪ k X ( δ + δ 2 ) ( log X ) B k . \int_{\mathbb{R}} \left|g\left(\frac{x}{X}\right)\right| \left|\sum_{n\asymp X} \Lambda^{*k}(n)\mathcal{K}_{\delta}(x+H-n)\right|^2 dx \ll_k X(\delta+\delta^2)(\log X)^{B_k}. ∫ R g ( X x ) n ≍ X ∑ Λ ∗ k ( n ) K δ ( x + H − n ) 2 d x ≪ k X ( δ + δ 2 ) ( log X ) B k . Collectively, this leads to
∫ R ∣ g ( x X ) ∣ ∣ Ψ k ( x ; H ) − 1 2 π i ∫ c − i T c + i T ( − ζ ′ ζ ( s ) ) k ( x + H ) s − x s s d s ∣ 2 d x ≪ k X ( δ + δ 2 ) ( log X ) B k . \int_{\mathbb{R}} \left|g\left(\frac{x}{X}\right)\right|\left|\Psi_k(x;H)-\frac{1}{2\pi i}\int_{c-iT}^{c+iT}\left(-\frac{\zeta'}{\zeta}(s)\right)^k\frac{(x+H)^s-x^s}{s}\,ds\right|^2 dx \ll_k X(\delta+\delta^2)(\log X)^{B_k}. ∫ R g ( X x ) Ψ k ( x ; H ) − 2 π i 1 ∫ c − i T c + i T ( − ζ ζ ′ ( s ) ) k s ( x + H ) s − x s d s 2 d x ≪ k X ( δ + δ 2 ) ( log X ) B k . Dividing through by H X HX H X δ = X T = H X − ϵ \delta=\frac{X}{T}=\sqrt{H}X^{-\epsilon} δ = T X = H X − ϵ
1 H X ∫ R ∣ g ( x X ) ∣ ∣ Ψ k ( x ; H ) − 1 2 π i ∫ c − i T c + i T ( − ζ ′ ζ ( s ) ) k ( x + H ) s − x s s d s ∣ 2 d x = o ( ( log X ) 2 k − 1 ) . \frac{1}{HX}\int_{\mathbb{R}} \left|g\left(\frac{x}{X}\right)\right|\left|\Psi_k(x;H)-\frac{1}{2\pi i}\int_{c-iT}^{c+iT}\left(-\frac{\zeta'}{\zeta}(s)\right)^k\frac{(x+H)^s-x^s}{s}\,ds\right|^2 dx=o\left((\log X)^{2k-1}\right). H X 1 ∫ R g ( X x ) Ψ k ( x ; H ) − 2 π i 1 ∫ c − i T c + i T ( − ζ ζ ′ ( s ) ) k s ( x + H ) s − x s d s 2 d x = o ( ( log X ) 2 k − 1 ) . We next shift from the line R e s = c \mathop{\mathrm{Re}}s=c Re s = c R e s = σ ∗ = 1 2 + 1 log T \mathop{\mathrm{Re}}s=\sigma_*=\frac{1}{2}+\frac{1}{\log T} Re s = σ ∗ = 2 1 + l o g T 1 − ζ ′ / ζ ( s ) -\zeta'/\zeta(s) − ζ ′ / ζ ( s ) ζ ( s ) \zeta(s) ζ ( s ) R e s = 1 / 2 \mathop{\mathrm{Re}}s=1/2 Re s = 1/2 s = 1 s=1 s = 1
1 2 π i ∫ c − i T c + i T ( − ζ ′ ζ ( s ) ) k ( x + H ) s − x s s d s = R e s s = 1 [ ( − ζ ′ ζ ( s ) ) k ( x + H ) s − x s s ] (199) \frac{1}{2\pi i}\int_{c-iT}^{c+iT}\left(-\frac{\zeta'}{\zeta}(s)\right)^k\frac{(x+H)^s-x^s}{s}\,ds=\mathop{\mathrm{Res}}_{s=1}\left[\left(-\frac{\zeta'}{\zeta}(s)\right)^k\frac{(x+H)^s-x^s}{s}\right] \tag*{(199)} 2 π i 1 ∫ c − i T c + i T ( − ζ ζ ′ ( s ) ) k s ( x + H ) s − x s d s = Res s = 1 [ ( − ζ ζ ′ ( s ) ) k s ( x + H ) s − x s ] (199) + P k ( x ; H ) + R h o r ( x ) , \qquad+\mathcal{P}_k(x;H)+R_{\mathrm{hor}}(x), + P k ( x ; H ) + R hor ( x ) , where
P k ( x ; H ) : = 1 2 π ∫ − T T ( − ζ ′ ζ ( σ ∗ + i t ) ) k ( x + H ) σ ∗ + i t − x σ ∗ + i t σ ∗ + i t d t , (200) \mathcal{P}_k(x;H):=\frac{1}{2\pi}\int_{-T}^{T}\left(-\frac{\zeta'}{\zeta}(\sigma_*+it)\right)^k\frac{(x+H)^{\sigma_*+it}-x^{\sigma_*+it}}{\sigma_*+it}\,dt, \tag*{(200)} P k ( x ; H ) := 2 π 1 ∫ − T T ( − ζ ζ ′ ( σ ∗ + i t ) ) k σ ∗ + i t ( x + H ) σ ∗ + i t − x σ ∗ + i t d t , (200) and
R h o r ( x ) : = 1 2 π i ( ∫ σ ∗ + i T c + i T + ∫ c − i T σ ∗ − i T ) ( − ζ ′ ζ ( s ) ) k ( x + H ) s − x s s d s . (201) R_{\mathrm{hor}}(x):=\frac{1}{2\pi i}\left(\int_{\sigma_*+iT}^{c+iT}+\int_{c-iT}^{\sigma_*-iT}\right)\left(-\frac{\zeta'}{\zeta}(s)\right)^k\frac{(x+H)^s-x^s}{s}\,ds. \tag*{(201)} R hor ( x ) := 2 π i 1 ( ∫ σ ∗ + i T c + i T + ∫ c − i T σ ∗ − i T ) ( − ζ ζ ′ ( s ) ) k s ( x + H ) s − x s d s . (201) We now bound the horizontal contribution R h o r R_{\mathrm{hor}} R hor s = σ ± i T s=\sigma\pm iT s = σ ± i T σ ∗ ≤ σ ≤ c \sigma_*\leq\sigma\leq c σ ∗ ≤ σ ≤ c (136) gives ζ ′ ( s ) / ζ ( s ) ≪ ( log T ) 2 \zeta'(s)/\zeta(s)\ll(\log T)^2 ζ ′ ( s ) / ζ ( s ) ≪ ( log T ) 2 x ≍ X x\asymp X x ≍ X
∣ ( x + H ) s − x s s ∣ ≪ min { H X σ − 1 , X σ T } . (202) \left|\frac{(x+H)^s-x^s}{s}\right|\ll\min\left\{HX^{\sigma-1},\frac{X^\sigma}{T}\right\}. \tag*{(202)} s ( x + H ) s − x s ≪ min { H X σ − 1 , T X σ } . (202) The first bound follows from ( x + H ) s − x s s = ∫ x x + H u s − 1 d u \frac{(x+H)^s-x^s}{s}=\int_x^{x+H}u^{s-1}\,du s ( x + H ) s − x s = ∫ x x + H u s − 1 d u ∣ s ∣ ≍ T |s|\asymp T ∣ s ∣ ≍ T X / ( T H ) = o ( 1 ) X/(TH)=o(1) X / ( T H ) = o ( 1 ) X X X
≪ ∫ σ ∗ c ( log T ) 2 k X σ T d σ ≪ X T ( log T ) 2 k + 1 . \ll\int_{\sigma_*}^{c}(\log T)^{2k}\frac{X^\sigma}{T}\,d\sigma\ll\frac{X}{T}(\log T)^{2k+1}. ≪ ∫ σ ∗ c ( log T ) 2 k T X σ d σ ≪ T X ( log T ) 2 k + 1 . Therefore
1 H X ∫ R g ( x X ) ∣ R h o r ( x ) ∣ 2 d x ≪ 1 H ( X T ) 2 ( log T ) 4 k + 2 = X − 2 ε ( log T ) 4 k + 2 = o ( ( log X ) 2 k − 1 ) . \frac{1}{HX}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)|R_{\mathrm{hor}}(x)|^2\,dx\ll\frac{1}{H}\left(\frac{X}{T}\right)^2(\log T)^{4k+2}=X^{-2\varepsilon}(\log T)^{4k+2}=o\left((\log X)^{2k-1}\right). H X 1 ∫ R g ( X x ) ∣ R hor ( x ) ∣ 2 d x ≪ H 1 ( T X ) 2 ( log T ) 4 k + 2 = X − 2 ε ( log T ) 4 k + 2 = o ( ( log X ) 2 k − 1 ) . The remainder R ( x ) R(x) R ( x ) R h o r ( x ) R_{\mathrm{hor}}(x) R hor ( x ) Ψ k \Psi_k Ψ k Δ k ( x ; H ) \Delta_k(x;H) Δ k ( x ; H ) (199) , we obtain
Δ k ( x ; H ) = P k ( x ; H ) + R ( x ) , (203) \Delta_k(x;H)=\mathcal{P}_k(x;H)+R(x), \tag*{(203)} Δ k ( x ; H ) = P k ( x ; H ) + R ( x ) , (203) where
1 H X ∫ R g ( x X ) ∣ R ( x ) ∣ 2 d x = o ( ( log X ) 2 k − 1 ) . (204) \frac{1}{HX}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)|R(x)|^2\,dx=o\left((\log X)^{2k-1}\right). \tag*{(204)} H X 1 ∫ R g ( X x ) ∣ R ( x ) ∣ 2 d x = o ( ( log X ) 2 k − 1 ) . (204) To pass from Δ k \Delta_k Δ k P k \mathcal{P}_k P k
∣ Δ k ∣ 2 = ∣ P k ∣ 2 + 2 ℜ ( P k R ‾ ) + ∣ R ∣ 2 . |\Delta_k|^2=|\mathcal{P}_k|^2+2\Re(\mathcal{P}_k\overline{R})+|R|^2. ∣ Δ k ∣ 2 = ∣ P k ∣ 2 + 2ℜ ( P k R ) + ∣ R ∣ 2 . Due to (204) , the term involving ∣ R ∣ 2 |R|^2 ∣ R ∣ 2
∣ 1 H X ∫ R g ( x X ) P k ( x ; H ) R ( x ) ‾ d x ∣ ≤ ( 1 H X ∫ R g ( x X ) ∣ P k ( x ; H ) ∣ 2 d x ) 1 / 2 ( 1 H X ∫ R g ( x X ) ∣ R ( x ) ∣ 2 d x ) 1 / 2 . (205) \begin{aligned}
\left|\frac{1}{HX}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)\mathcal{P}_k(x;H)\overline{R(x)}\,dx\right| \\
&\le\left(\frac{1}{HX}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)|\mathcal{P}_k(x;H)|^2\,dx\right)^{1/2}\left(\frac{1}{HX}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)|R(x)|^2\,dx\right)^{1/2}. \tag*{(205)}
\end{aligned} H X 1 ∫ R g ( X x ) P k ( x ; H ) R ( x ) d x ≤ ( H X 1 ∫ R g ( X x ) ∣ P k ( x ; H ) ∣ 2 d x ) 1/2 ( H X 1 ∫ R g ( X x ) ∣ R ( x ) ∣ 2 d x ) 1/2 . (205) We now turn to the mean square of P k \mathcal{P}_k P k V ~ g , k ( X , H ) \widetilde{V}_{g,k}(X,H) V g , k ( X , H )
1 H X ∫ R g ( x X ) ∣ P k ( x ; H ) ∣ 2 d x ≪ ( log X ) 2 k − 1 , \frac{1}{HX}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)|\mathcal{P}_k(x;H)|^2\,dx\ll(\log X)^{2k-1}, H X 1 ∫ R g ( X x ) ∣ P k ( x ; H ) ∣ 2 d x ≪ ( log X ) 2 k − 1 , needed to dispose of the cross-term in (205) . Once this has been proven, the replacement of Δ k \Delta_k Δ k P k \mathcal{P}_k P k (204) . Thus, the reduction we aim to prove is
V ~ g , k ( X , H ) = 1 H X ∫ R g ( x X ) ∣ 1 2 π ∫ − T T ( − ζ ′ ζ ( σ ∗ + i t ) ) k ( x + H ) σ ∗ + i t − x σ ∗ + i t σ ∗ + i t d t ∣ 2 d x \widetilde{V}_{g,k}(X,H)=\frac{1}{HX}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)\left|\frac{1}{2\pi}\int_{-T}^{T}\left(-\frac{\zeta'}{\zeta}(\sigma_*+it)\right)^k\frac{(x+H)^{\sigma_*+it}-x^{\sigma_*+it}}{\sigma_*+it}\,dt\right|^2dx V g , k ( X , H ) = H X 1 ∫ R g ( X x ) 2 π 1 ∫ − T T ( − ζ ζ ′ ( σ ∗ + i t ) ) k σ ∗ + i t ( x + H ) σ ∗ + i t − x σ ∗ + i t d t 2 d x + o ( ( log X ) 2 k − 1 ) . (206) \qquad+o\left((\log X)^{2k-1}\right). \tag*{(206)} + o ( ( log X ) 2 k − 1 ) . (206) To isolate the main contribution to the mean square in (206) , set
Z : = ( log log X ) A , A > 2. Z:=(\log\log X)^A,\qquad A>2. Z := ( log log X ) A , A > 2. We shall show that, after normalisation by H X HX H X t t t X H Z ≤ ∣ t ∣ ≤ X Z H \frac{X}{HZ}\le|t|\le\frac{XZ}{H} H Z X ≤ ∣ t ∣ ≤ H X Z
E 1 : = [ 0 , X H Z ] , E 2 : = [ X Z H , X Z H log T ] , E 3 : = [ X Z H log T , T ] . E_1:=\left[0,\frac{X}{HZ}\right],\qquad E_2:=\left[\frac{XZ}{H},\frac{XZ}{H}\log T\right],\qquad E_3:=\left[\frac{XZ}{H}\log T,T\right]. E 1 := [ 0 , H Z X ] , E 2 := [ H X Z , H X Z log T ] , E 3 := [ H X Z log T , T ] . The choice of ε \varepsilon ε
( Z / h ) log T T = Z log T X ε H = o ( 1 ) , \frac{(Z/h)\log T}{T}=\frac{Z\log T}{X^{\varepsilon}\sqrt{H}}=o(1), T ( Z / h ) log T = X ε H Z log T = o ( 1 ) , so these ranges are contained in [ 0 , T ] [0,T] [ 0 , T ] X X X
1 H X ∫ R g ( x X ) ∣ 1 2 π ∫ ∣ t ∣ ∈ E 1 ∪ E 2 ∪ E 3 ( − ζ ′ ζ ( σ ∗ + i t ) ) k ( x + H ) σ ∗ + i t − x σ ∗ + i t σ ∗ + i t d t ∣ 2 d x = o ( ( log X ) 2 k − 1 ) . (207) \begin{aligned}
\frac{1}{HX}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)\left|\frac{1}{2\pi}\int_{|t|\in E_1\cup E_2\cup E_3}\left(-\frac{\zeta'}{\zeta}(\sigma_*+it)\right)^k\frac{(x+H)^{\sigma_*+it}-x^{\sigma_*+it}}{\sigma_*+it}\,dt\right|^2dx \\
&=o\left((\log X)^{2k-1}\right). \tag*{(207)}
\end{aligned} H X 1 ∫ R g ( X x ) 2 π 1 ∫ ∣ t ∣ ∈ E 1 ∪ E 2 ∪ E 3 ( − ζ ζ ′ ( σ ∗ + i t ) ) k σ ∗ + i t ( x + H ) σ ∗ + i t − x σ ∗ + i t d t 2 d x = o ( ( log X ) 2 k − 1 ) . (207) Using a Saffari–Vaughan-type reduction, i.e. proceeding as in [29 Page 25] (see also [26 Page 5] , [3 Page 12] ), it is enough for (207) to prove
X 2 / log T H ∫ R g ( x X ) ∣ 1 2 π ∫ ∣ t ∣ ∈ E 1 ∪ E 2 ∪ E 3 ( − ζ ′ ζ ( σ ∗ + i t ) ) k x i t ( 1 + u ) σ ∗ + i t − 1 σ ∗ + i t d t ∣ 2 d x = o ( ( log X ) 2 k − 1 ) (208) \frac{X^{2/\log T}}{H}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)\left|\frac{1}{2\pi}\int_{|t|\in E_{1}\cup E_{2}\cup E_{3}}\left(-\frac{\zeta'}{\zeta}(\sigma_{*}+it)\right)^{k}x^{it}\frac{(1+u)^{\sigma_{*}+it}-1}{\sigma_{*}+it}\,dt\right|^{2}dx
= o\left((\log X)^{2k-1}\right) \tag*{(208)} H X 2/ l o g T ∫ R g ( X x ) 2 π 1 ∫ ∣ t ∣ ∈ E 1 ∪ E 2 ∪ E 3 ( − ζ ζ ′ ( σ ∗ + i t ) ) k x i t σ ∗ + i t ( 1 + u ) σ ∗ + i t − 1 d t 2 d x = o ( ( log X ) 2 k − 1 ) (208) uniformly for all u ≪ H / X u \ll H/X u ≪ H / X ∣ t ∣ ∈ [ 0 , X H Z ] ∪ [ X Z H , T ] |t| \in\left[0,\frac{X}{HZ}\right]\cup\left[\frac{XZ}{H},T\right] ∣ t ∣ ∈ [ 0 , H Z X ] ∪ [ H X Z , T ] ∣ t ∣ ∈ [ X H Z , X Z H ] |t| \in\left[\frac{X}{HZ},\frac{XZ}{H}\right] ∣ t ∣ ∈ [ H Z X , H X Z ] § 3.10 , we will continue with the exact kernel. Expanding the squared modulus in (208) gives a double integral in variables t 1 , t 2 t_{1},t_{2} t 1 , t 2 s = t 1 − t 2 s=t_{1}-t_{2} s = t 1 − t 2 z = t 1 z=t_{1} z = t 1
X 1 + 2 / log T H ( 2 π ) 2 ∭ ∣ z ∣ , ∣ z − s ∣ ∈ E 1 ∪ E 2 ∪ E 3 X i s g ^ ( 1 + i s ) ( − ζ ′ ζ ( σ ∗ + i z ) ) k ( − ζ ′ ζ ( σ ∗ + i ( s − z ) ) ) k ( ( 1 + u ) σ ∗ + i z − 1 ) ( ( 1 + u ) σ ∗ + i ( s − z ) − 1 ) ( σ ∗ + i z ) ( σ ∗ + i ( s − z ) ) d z d s . (209) \frac{X^{1+2/\log T}}{H(2\pi)^{2}}\iiint_{|z|,\,|z-s|\in E_{1}\cup E_{2}\cup E_{3}}X^{is}\widehat{g}(1+is)
\left(-\frac{\zeta'}{\zeta}(\sigma_{*}+iz)\right)^{k}
\left(-\frac{\zeta'}{\zeta}(\sigma_{*}+i(s-z))\right)^{k}
\frac{\left((1+u)^{\sigma_{*}+iz}-1\right)\left((1+u)^{\sigma_{*}+i(s-z)}-1\right)}{(\sigma_{*}+iz)(\sigma_{*}+i(s-z))}\,dz\,ds. \tag*{(209)} H ( 2 π ) 2 X 1 + 2/ l o g T ∭ ∣ z ∣ , ∣ z − s ∣ ∈ E 1 ∪ E 2 ∪ E 3 X i s g ( 1 + i s ) ( − ζ ζ ′ ( σ ∗ + i z ) ) k ( − ζ ζ ′ ( σ ∗ + i ( s − z )) ) k ( σ ∗ + i z ) ( σ ∗ + i ( s − z )) ( ( 1 + u ) σ ∗ + i z − 1 ) ( ( 1 + u ) σ ∗ + i ( s − z ) − 1 ) d z d s . (209) The x x x g g g
g ^ ( w ) : = ∫ 0 ∞ g ( v ) v w − 1 d v . (210) \widehat{g}(w):=\int_{0}^{\infty}g(v)v^{w-1}\,dv. \tag*{(210)} g ( w ) := ∫ 0 ∞ g ( v ) v w − 1 d v . (210) After the change of variables x = X v x=Xv x = X v
∫ R g ( x X ) x i s d x = X 1 + i s g ^ ( 1 + i s ) , \int_{\mathbb{R}}g\left(\frac{x}{X}\right)x^{is}\,dx=X^{1+is}\widehat{g}(1+is), ∫ R g ( X x ) x i s d x = X 1 + i s g ( 1 + i s ) , the difference variable s = t 1 − t 2 s=t_{1}-t_{2} s = t 1 − t 2 g ^ ( 1 + i s ) \widehat{g}(1+is) g ( 1 + i s ) (209) . Define h : = H X h:=\frac{H}{X} h := X H
F ( z , s , u ) : = ( ( 1 + u ) σ ∗ + i z − 1 ) ( ( 1 + u ) σ ∗ − i z + i s − 1 ) . (211) F(z,s,u):=\left((1+u)^{\sigma_{*}+iz}-1\right)\left((1+u)^{\sigma_{*}-iz+is}-1\right). \tag*{(211)} F ( z , s , u ) := ( ( 1 + u ) σ ∗ + i z − 1 ) ( ( 1 + u ) σ ∗ − i z + i s − 1 ) . (211) The identity ( 1 + u ) w − 1 = w ∫ 0 u ( 1 + r ) w − 1 d r (1+u)^{w}-1=w\int_{0}^{u}(1+r)^{w-1}\,dr ( 1 + u ) w − 1 = w ∫ 0 u ( 1 + r ) w − 1 d r
∣ ( 1 + u ) w − 1 w ∣ ≤ ∫ 0 u ( 1 + r ) σ ∗ − 1 d r ≪ u . \left|\frac{(1+u)^{w}-1}{w}\right|\leq\int_{0}^{u}(1+r)^{\sigma_{*}-1}\,dr\ll u. w ( 1 + u ) w − 1 ≤ ∫ 0 u ( 1 + r ) σ ∗ − 1 d r ≪ u . On the other hand, we have
∣ ( 1 + u ) w − 1 w ∣ ≤ ( 1 + u ) σ ∗ + 1 ∣ w ∣ ≪ ∣ w ∣ − 1 , \left|\frac{(1+u)^{w}-1}{w}\right|\leq\frac{(1+u)^{\sigma_{*}}+1}{|w|}\ll|w|^{-1}, w ( 1 + u ) w − 1 ≤ ∣ w ∣ ( 1 + u ) σ ∗ + 1 ≪ ∣ w ∣ − 1 , because u ≪ h = o ( 1 ) u\ll h=o(1) u ≪ h = o ( 1 ) σ ∗ \sigma_{*} σ ∗
∣ ( 1 + u ) w − 1 w ∣ ≪ min ( u , ∣ w ∣ − 1 ) , 0 ≤ u ≪ h , \left|\frac{(1+u)^{w}-1}{w}\right|\ll\min(u,|w|^{-1}),\qquad0\leq u\ll h, w ( 1 + u ) w − 1 ≪ min ( u , ∣ w ∣ − 1 ) , 0 ≤ u ≪ h , uniformly over the w w w a = σ ∗ + i z a=\sigma_{*}+iz a = σ ∗ + i z b = σ ∗ − i z + i s b=\sigma_{*}-iz+is b = σ ∗ − i z + i s
∣ ζ ′ ζ ( a ) ∣ k ∣ ζ ′ ζ ( b ) ∣ k ∣ F ( z , s , u ) a b ∣ ≪ ∣ ζ ′ ζ ( a ) ∣ k ∣ ζ ′ ζ ( b ) ∣ k min ( u , ∣ a ∣ − 1 ) min ( u , ∣ b ∣ − 1 ) ≪ ∣ ζ ′ ζ ( a ) ∣ 2 k min ( u 2 , ∣ a ∣ − 2 ) + ∣ ζ ′ ζ ( b ) ∣ 2 k min ( u 2 , ∣ b ∣ − 2 ) , (212) \begin{aligned}
\left|\frac{\zeta'}{\zeta}(a)\right|^{k}\left|\frac{\zeta'}{\zeta}(b)\right|^{k}\left|\frac{F(z,s,u)}{ab}\right|
\ll\left|\frac{\zeta'}{\zeta}(a)\right|^{k}\left|\frac{\zeta'}{\zeta}(b)\right|^{k}\min(u,|a|^{-1})\min(u,|b|^{-1}) \\
&\ll\left|\frac{\zeta'}{\zeta}(a)\right|^{2k}\min(u^{2},|a|^{-2})+\left|\frac{\zeta'}{\zeta}(b)\right|^{2k}\min(u^{2},|b|^{-2}), \tag*{(212)}
\end{aligned} ζ ζ ′ ( a ) k ζ ζ ′ ( b ) k ab F ( z , s , u ) ≪ ζ ζ ′ ( a ) k ζ ζ ′ ( b ) k min ( u , ∣ a ∣ − 1 ) min ( u , ∣ b ∣ − 1 ) ≪ ζ ζ ′ ( a ) 2 k min ( u 2 , ∣ a ∣ − 2 ) + ζ ζ ′ ( b ) 2 k min ( u 2 , ∣ b ∣ − 2 ) , (212) where we used A B ≤ ( A 2 + B 2 ) / 2 AB\leq(A^{2}+B^{2})/2 A B ≤ ( A 2 + B 2 ) /2 E 1 E_{1} E 1 E 2 E_{2} E 2 E 3 E_{3} E 3 (209) , we first restrict this integral to cases in which t 1 t_1 t 1 t 2 t_2 t 2 (210) makes pairs with large ∣ t 1 − t 2 ∣ \lvert t_1-t_2\rvert ∣ t 1 − t 2 ∣ η > 0 \eta>0 η > 0
S : = ( log X ) η . S := (\log X)^\eta. S := ( log X ) η . We now estimate the portion with ∣ s ∣ > S \lvert s\rvert>S ∣ s ∣ > S ∣ s ∣ ≤ S \lvert s\rvert\le S ∣ s ∣ ≤ S
Bounds on tail regions
3.9. Bounds on tail regions. Taking absolute values in (209) and applying (212) , the contribution from ∣ s ∣ > S \lvert s\rvert>S ∣ s ∣ > S (212) . For the second term, we make the substitution y = s − z y=s-z y = s − z s s s
∣ z ∣ ∈ E 1 ∪ E 2 ∪ E 3 , ∣ z − s ∣ ∈ E 1 ∪ E 2 ∪ E 3 , \lvert z\rvert\in E_1\cup E_2\cup E_3,\qquad\lvert z-s\rvert\in E_1\cup E_2\cup E_3, ∣ z ∣ ∈ E 1 ∪ E 2 ∪ E 3 , ∣ z − s ∣ ∈ E 1 ∪ E 2 ∪ E 3 , from (209) and transforms the second term into the first. For each fixed s s s d z dz d z z z z ∣ s ∣ > S \lvert s\rvert>S ∣ s ∣ > S (209) is twice either contribution. Since the resulting integrand is non-negative, we may omit the condition ∣ z − s ∣ ∈ E 1 ∪ E 2 ∪ E 3 \lvert z-s\rvert\in E_1\cup E_2\cup E_3 ∣ z − s ∣ ∈ E 1 ∪ E 2 ∪ E 3 z z z ∣ z ∣ ≤ T \lvert z\rvert\le T ∣ z ∣ ≤ T 2 2 2
X 1 + 2 / log T H ≍ 1 h , \frac{X^{1+2/\log T}}{H}\asymp\frac{1}{h}, H X 1 + 2/ l o g T ≍ h 1 , it is sufficient to prove
1 h ∫ ∣ s ∣ > S ∣ g ^ ( 1 + i s ) ∣ ∫ ∣ z ∣ ≤ T ∣ ζ ′ ζ ( σ ∗ + i z ) ∣ 2 k min ( u 2 , ∣ σ ∗ + i z ∣ − 2 ) d z d s = o ( ( log X ) 2 k − 1 ) . (213) \frac{1}{h}\int_{\lvert s\rvert>S}\lvert\widehat{g}(1+is)\rvert\int_{\lvert z\rvert\le T}\left\lvert\frac{\zeta'}{\zeta}(\sigma_*+iz)\right\rvert^{2k}\min\left(u^2,\lvert\sigma_*+iz\rvert^{-2}\right)\,dz\,ds=o\left((\log X)^{2k-1}\right). \tag*{(213)} h 1 ∫ ∣ s ∣ > S ∣ g ( 1 + i s )∣ ∫ ∣ z ∣ ≤ T ζ ζ ′ ( σ ∗ + i z ) 2 k min ( u 2 , ∣ σ ∗ + i z ∣ − 2 ) d z d s = o ( ( log X ) 2 k − 1 ) . (213) As g ∈ C ∞ ( ( 0 , ∞ ) ) g\in C^\infty((0,\infty)) g ∈ C ∞ (( 0 , ∞ )) (210) gives
∣ g ^ ( 1 + i s ) ∣ ≪ B , g ( 1 + ∣ s ∣ ) − B (214) \lvert\widehat{g}(1+is)\rvert\ll_{B,g}(1+\lvert s\rvert)^{-B} \tag*{(214)} ∣ g ( 1 + i s )∣ ≪ B , g ( 1 + ∣ s ∣ ) − B (214) for every B > 0 B>0 B > 0 B > 1 B>1 B > 1
∫ R ∣ g ^ ( 1 + i s ) ∣ d s ≪ g 1 , (215) \int_{\mathbb{R}}\lvert\widehat{g}(1+is)\rvert\,ds\ll_g 1, \tag*{(215)} ∫ R ∣ g ( 1 + i s )∣ d s ≪ g 1 , (215) and
∫ ∣ s ∣ > S ∣ g ^ ( 1 + i s ) ∣ d s ≪ B , g S 1 − B . (216) \int_{\lvert s\rvert>S}\lvert\widehat{g}(1+is)\rvert\,ds\ll_{B,g}S^{1-B}. \tag*{(216)} ∫ ∣ s ∣ > S ∣ g ( 1 + i s )∣ d s ≪ B , g S 1 − B . (216) We first dispose of the bounded range ∣ z ∣ < 1 \lvert z\rvert<1 ∣ z ∣ < 1 σ ∗ = 1 2 + 1 log T \sigma_*=\frac{1}{2}+\frac{1}{\log T} σ ∗ = 2 1 + l o g T 1 T T T 1 2 < σ ∗ ≤ 3 4 \frac{1}{2}<\sigma_*\le\frac{3}{4} 2 1 < σ ∗ ≤ 4 3 z z z ∣ z ∣ < 1 \lvert z\rvert<1 ∣ z ∣ < 1 σ ∗ + i z \sigma_*+iz σ ∗ + i z
{ w ∈ C : 1 2 ≤ ℜ w ≤ 3 4 , ∣ ℑ w ∣ ≤ 1 } . \left\{w\in\mathbb{C}:\frac{1}{2}\le\Re w\le\frac{3}{4},\ \lvert\Im w\rvert\le1\right\}. { w ∈ C : 2 1 ≤ ℜ w ≤ 4 3 , ∣ ℑ w ∣ ≤ 1 } . As ζ ( w ) \zeta(w) ζ ( w ) ζ ′ / ζ \zeta'/\zeta ζ ′ / ζ
ζ ′ ζ ( σ ∗ + i z ) ≪ 1 , ∣ z ∣ < 1 , (217) \frac{\zeta'}{\zeta}(\sigma_*+iz)\ll1,\qquad\lvert z\rvert<1, \tag*{(217)} ζ ζ ′ ( σ ∗ + i z ) ≪ 1 , ∣ z ∣ < 1 , (217) with an implied constant independent of T T T min ( u 2 , ∣ σ ∗ + i z ∣ − 2 ) ≤ u 2 \min(u^2,\lvert\sigma_*+iz\rvert^{-2})\le u^2 min ( u 2 , ∣ σ ∗ + i z ∣ − 2 ) ≤ u 2 (216) gives
1 h ∫ ∣ s ∣ > S ∣ g ^ ( 1 + i s ) ∣ ∫ ∣ z ∣ < 1 ∣ ζ ′ ζ ( σ ∗ + i z ) ∣ 2 k min ( u 2 , ∣ σ ∗ + i z ∣ − 2 ) d z d s ≪ B , g u 2 h S 1 − B ≪ B , g h S 1 − B , (218) \frac{1}{h}\int_{\lvert s\rvert>S}\lvert\widehat{g}(1+is)\rvert\int_{\lvert z\rvert<1}\left\lvert\frac{\zeta'}{\zeta}(\sigma_*+iz)\right\rvert^{2k}\min\left(u^2,\lvert\sigma_*+iz\rvert^{-2}\right)\,dz\,ds
\ll_{B,g}\frac{u^2}{h}S^{1-B}\ll_{B,g}hS^{1-B}, \tag*{(218)} h 1 ∫ ∣ s ∣ > S ∣ g ( 1 + i s )∣ ∫ ∣ z ∣ < 1 ζ ζ ′ ( σ ∗ + i z ) 2 k min ( u 2 , ∣ σ ∗ + i z ∣ − 2 ) d z d s ≪ B , g h u 2 S 1 − B ≪ B , g h S 1 − B , (218) where u ≪ h u \ll h u ≪ h u 2 / h ≪ h u^2/h \ll h u 2 / h ≪ h h = X − 1 / α h=X^{-1/\alpha} h = X − 1/ α S = ( log X ) η S=(\log X)^\eta S = ( log X ) η o ( ( log X ) 2 k − 1 ) o((\log X)^{2k-1}) o (( log X ) 2 k − 1 ) 1 ≤ ∣ z ∣ ≤ T 1\leq|z|\leq T 1 ≤ ∣ z ∣ ≤ T (213) is even in the real variable z z z
∣ ζ ′ ζ ( σ ∗ − i z ) ∣ = ∣ ζ ′ ζ ( σ ∗ + i z ) ∣ , ∣ σ ∗ − i z ∣ = ∣ σ ∗ + i z ∣ . \left|\frac{\zeta'}{\zeta}(\sigma_* - iz)\right|=\left|\frac{\zeta'}{\zeta}(\sigma_* + iz)\right|,\qquad|\sigma_* - iz|=|\sigma_*+iz|. ζ ζ ′ ( σ ∗ − i z ) = ζ ζ ′ ( σ ∗ + i z ) , ∣ σ ∗ − i z ∣ = ∣ σ ∗ + i z ∣. Thus the positive and negative z z z z ≥ 1 z\geq1 z ≥ 1 ∣ σ ∗ + i z ∣ − 2 ≤ z − 2 |\sigma_*+iz|^{-2}\leq z^{-2} ∣ σ ∗ + i z ∣ − 2 ≤ z − 2
∫ 1 T ∣ ζ ′ ζ ( σ ∗ + i z ) ∣ 2 k min ( u 2 , z − 2 ) d z . (219) \int_1^T\left|\frac{\zeta'}{\zeta}(\sigma_*+iz)\right|^{2k}\min(u^2,z^{-2})\,dz. \tag*{(219)} ∫ 1 T ζ ζ ′ ( σ ∗ + i z ) 2 k min ( u 2 , z − 2 ) d z . (219) Define the cumulative moment
A ( Y ) : = ∫ 1 Y ∣ ζ ′ ζ ( σ ∗ + i z ) ∣ 2 k d z . (220) A(Y):=\int_1^Y\left|\frac{\zeta'}{\zeta}(\sigma_*+iz)\right|^{2k}\,dz. \tag*{(220)} A ( Y ) := ∫ 1 Y ζ ζ ′ ( σ ∗ + i z ) 2 k d z . (220) We claim that, uniformly for 1 ≤ Y ≤ T 1\leq Y\leq T 1 ≤ Y ≤ T
A ( Y ) ≪ Y ( log X ) 2 k . (221) A(Y)\ll Y(\log X)^{2k}. \tag*{(221)} A ( Y ) ≪ Y ( log X ) 2 k . (221) The bound (42) from Proposition 2.9 holds once its height parameter is sufficiently large. We therefore fix a constant Y 0 ≥ 2 Y_0\geq2 Y 0 ≥ 2 X X X T T T Y Y Y (42) is valid whenever its averaging height is at least Y 0 Y_0 Y 0 Y 0 ≤ Y ≤ T Y_0\leq Y\leq T Y 0 ≤ Y ≤ T (42) with averaging height Y Y Y
z ~ 1 = σ ∗ − 1 2 = 1 log T , z ~ 2 = 1 2 − σ ∗ = − 1 log T . \widetilde z_1=\sigma_*-\frac{1}{2}=\frac{1}{\log T},\qquad\widetilde z_2=\frac{1}{2}-\sigma_*=-\frac{1}{\log T}. z 1 = σ ∗ − 2 1 = log T 1 , z 2 = 2 1 − σ ∗ = − log T 1 . Here, applying the proposition with averaging height Y Y Y Y Y Y T T T σ ∗ \sigma_* σ ∗ Y ≤ T Y\leq T Y ≤ T
0 < − R e z ~ 2 = 1 log T ≤ 3 log Y , 0 < R e ( z ~ 1 − z ~ 2 ) = 2 log T ≤ 3 log Y . 0<-\mathop{\rm Re}\widetilde z_2=\frac{1}{\log T}\leq\frac{3}{\log Y},\qquad0<\mathop{\rm Re}(\widetilde z_1-\widetilde z_2)=\frac{2}{\log T}\leq\frac{3}{\log Y}. 0 < − Re z 2 = log T 1 ≤ log Y 3 , 0 < Re ( z 1 − z 2 ) = log T 2 ≤ log Y 3 . The non-negativity of the integrand in (220) now gives
A ( Y ) ≤ ∫ 0 Y ∣ ζ ′ ζ ( σ ∗ + i z ) ∣ 2 k d z ≪ Y ( log Y + 1 2 log T ) 2 k ≪ Y ( log X ) 2 k , A(Y)\leq\int_0^Y\left|\frac{\zeta'}{\zeta}(\sigma_*+iz)\right|^{2k}dz\ll Y\left(\log Y+\frac{1}{2}\log T\right)^{2k}\ll Y(\log X)^{2k}, A ( Y ) ≤ ∫ 0 Y ζ ζ ′ ( σ ∗ + i z ) 2 k d z ≪ Y ( log Y + 2 1 log T ) 2 k ≪ Y ( log X ) 2 k , where we use Y ≤ T Y\leq T Y ≤ T log T ≍ log X \log T\asymp\log X log T ≍ log X 1 ≤ Y ≤ Y 0 1\leq Y\leq Y_0 1 ≤ Y ≤ Y 0 ζ ′ / ζ \zeta'/\zeta ζ ′ / ζ 1 ≤ z ≤ Y 0 1\leq z\leq Y_0 1 ≤ z ≤ Y 0
∣ ζ ′ ζ ( σ ∗ + i z ) ∣ ≪ Y 0 1 + ∑ ρ = 1 2 + i γ ∣ γ − z ∣ ≤ 1 1 ∣ σ ∗ + i z − ρ ∣ ≪ Y 0 log T . \left|\frac{\zeta'}{\zeta}(\sigma_*+iz)\right|\ll_{Y_0}1+\sum_{\substack{\rho=\frac{1}{2}+i\gamma\\|\gamma-z|\leq1}}\frac{1}{|\sigma_*+iz-\rho|}\ll_{Y_0}\log T. ζ ζ ′ ( σ ∗ + i z ) ≪ Y 0 1 + ρ = 2 1 + iγ ∣ γ − z ∣ ≤ 1 ∑ ∣ σ ∗ + i z − ρ ∣ 1 ≪ Y 0 log T . Only finitely many zeros occur in this fixed bounded range, while RH gives ∣ σ ∗ + i z − ρ ∣ ≥ σ ∗ − 1 2 = 1 log T |\sigma_*+iz-\rho|\geq\sigma_*-\frac{1}{2}=\frac{1}{\log T} ∣ σ ∗ + i z − ρ ∣ ≥ σ ∗ − 2 1 = l o g T 1 1 ≤ Y ≤ Y 0 1\leq Y\leq Y_0 1 ≤ Y ≤ Y 0
A ( Y ) ≪ k , Y 0 Y ( log T ) 2 k ≪ k , Y 0 Y ( log X ) 2 k . A(Y)\ll_{k,Y_0}Y(\log T)^{2k}\ll_{k,Y_0}Y(\log X)^{2k}. A ( Y ) ≪ k , Y 0 Y ( log T ) 2 k ≪ k , Y 0 Y ( log X ) 2 k . The constant Y 0 Y_0 Y 0 (221) throughout 1 ≤ Y ≤ T 1\leq Y\leq T 1 ≤ Y ≤ T
By Stieltjes integration, we write (219) as
∫ 1 T ∣ ζ ′ ζ ( σ ∗ + i z ) ∣ 2 k min ( u 2 , z − 2 ) d z = ∫ 1 T min ( u 2 , z − 2 ) d A ( z ) . (222) \int_1^T\left|\frac{\zeta'}{\zeta}(\sigma_*+iz)\right|^{2k}\min(u^2,z^{-2})\,dz=\int_1^T\min(u^2,z^{-2})\,dA(z). \tag*{(222)} ∫ 1 T ζ ζ ′ ( σ ∗ + i z ) 2 k min ( u 2 , z − 2 ) d z = ∫ 1 T min ( u 2 , z − 2 ) d A ( z ) . (222) The case u = 0 u=0 u = 0 u > 0 u>0 u > 0 u ≪ h = X − 1 / α u\ll h=X^{-1/\alpha} u ≪ h = X − 1/ α u < 1 u<1 u < 1 X X X 1 / u > 1 1/u>1 1/ u > 1 z = 1 / u z=1/u z = 1/ u 1 / u ≤ T 1/u\le T 1/ u ≤ T
∫ 1 T min ( u 2 , z − 2 ) d A ( z ) = u 2 A ( 1 / u ) + ∫ 1 / u T z − 2 d A ( z ) . \int_1^T \min(u^2,z^{-2})\,dA(z)=u^2A(1/u)+\int_{1/u}^T z^{-2}\,dA(z). ∫ 1 T min ( u 2 , z − 2 ) d A ( z ) = u 2 A ( 1/ u ) + ∫ 1/ u T z − 2 d A ( z ) . By the bound in (221) , u 2 A ( 1 / u ) ≪ u ( log X ) 2 k u^2A(1/u)\ll u(\log X)^{2k} u 2 A ( 1/ u ) ≪ u ( log X ) 2 k A ( z ) A(z) A ( z )
∫ 1 / u T z − 2 d A ( z ) ≪ T − 2 A ( T ) + u 2 A ( 1 / u ) + ∫ 1 / u T A ( z ) z − 3 d z ≪ u ( log X ) 2 k , \int_{1/u}^T z^{-2}\,dA(z)\ll T^{-2}A(T)+u^2A(1/u)+\int_{1/u}^T A(z)z^{-3}\,dz\ll u(\log X)^{2k}, ∫ 1/ u T z − 2 d A ( z ) ≪ T − 2 A ( T ) + u 2 A ( 1/ u ) + ∫ 1/ u T A ( z ) z − 3 d z ≪ u ( log X ) 2 k , using T − 1 ≤ u T^{-1}\le u T − 1 ≤ u ∫ 1 / u T z − 2 d z ≪ u \int_{1/u}^T z^{-2}\,dz\ll u ∫ 1/ u T z − 2 d z ≪ u 1 / u > T 1/u>T 1/ u > T min ( u 2 , z − 2 ) = u 2 \min(u^2,z^{-2})=u^2 min ( u 2 , z − 2 ) = u 2 1 ≤ z ≤ T 1\le z\le T 1 ≤ z ≤ T
∫ 1 T min ( u 2 , z − 2 ) d A ( z ) = u 2 A ( T ) ≪ u 2 T ( log X ) 2 k ≪ u ( log X ) 2 k . \int_1^T \min(u^2,z^{-2})\,dA(z)=u^2A(T)\ll u^2T(\log X)^{2k}\ll u(\log X)^{2k}. ∫ 1 T min ( u 2 , z − 2 ) d A ( z ) = u 2 A ( T ) ≪ u 2 T ( log X ) 2 k ≪ u ( log X ) 2 k . As u ≪ h u\ll h u ≪ h
∫ 1 T ∣ ζ ′ ζ ( σ ∗ + i z ) ∣ 2 k min ( u 2 , z − 2 ) d z ≪ h ( log X ) 2 k . (223) \int_1^T \left|\frac{\zeta'}{\zeta}(\sigma_*+iz)\right|^{2k}\min(u^2,z^{-2})\,dz\ll h(\log X)^{2k}. \tag*{(223)} ∫ 1 T ζ ζ ′ ( σ ∗ + i z ) 2 k min ( u 2 , z − 2 ) d z ≪ h ( log X ) 2 k . (223) Applying (216) and (223) to the part of (213) with 1 ≤ ∣ z ∣ ≤ T 1\le|z|\le T 1 ≤ ∣ z ∣ ≤ T
1 h ∫ ∣ s ∣ > S ∣ g ^ ( 1 + i s ) ∣ ∫ 1 ≤ ∣ z ∣ ≤ T ∣ ζ ′ ζ ( σ ∗ + i z ) ∣ 2 k min ( u 2 , ∣ σ ∗ + i z ∣ − 2 ) d z d s ≪ B , g ( log X ) 2 k S 1 − B . \frac{1}{h}\int_{|s|>S}|\widehat{g}(1+is)|\int_{1\le|z|\le T}\left|\frac{\zeta'}{\zeta}(\sigma_*+iz)\right|^{2k}\min(u^2,|\sigma_*+iz|^{-2})\,dz\,ds
\ll_{B,g}(\log X)^{2k}S^{1-B}. h 1 ∫ ∣ s ∣ > S ∣ g ( 1 + i s ) ∣ ∫ 1 ≤ ∣ z ∣ ≤ T ζ ζ ′ ( σ ∗ + i z ) 2 k min ( u 2 , ∣ σ ∗ + i z ∣ − 2 ) d z d s ≪ B , g ( log X ) 2 k S 1 − B . Together with the bounded-z z z (218) , this shows that the full expression in (213) is
≪ B , g h S 1 − B + ( log X ) 2 k S 1 − B ≪ B , g ( log X ) 2 k S 1 − B . \ll_{B,g}hS^{1-B}+(\log X)^{2k}S^{1-B}\ll_{B,g}(\log X)^{2k}S^{1-B}. ≪ B , g h S 1 − B + ( log X ) 2 k S 1 − B ≪ B , g ( log X ) 2 k S 1 − B . Selecting B B B η ( B − 1 ) > 1 \eta(B-1)>1 η ( B − 1 ) > 1 S = ( log X ) η S=(\log X)^\eta S = ( log X ) η
( log X ) 2 k S 1 − B = ( log X ) 2 k − η ( B − 1 ) = o ( ( log X ) 2 k − 1 ) . (\log X)^{2k}S^{1-B}=(\log X)^{2k-\eta(B-1)}=o\left((\log X)^{2k-1}\right). ( log X ) 2 k S 1 − B = ( log X ) 2 k − η ( B − 1 ) = o ( ( log X ) 2 k − 1 ) . This proves (213) . Hence, when estimating the contributions from E 1 E_1 E 1 E 2 E_2 E 2 E 3 E_3 E 3 ∣ s ∣ ≤ S = ( log X ) η |s|\le S=(\log X)^\eta ∣ s ∣ ≤ S = ( log X ) η s s s z z z E j E_j E j
∣ z − s ∣ ∈ E 1 ∪ E 2 ∪ E 3 |z-s|\in E_1\cup E_2\cup E_3 ∣ z − s ∣ ∈ E 1 ∪ E 2 ∪ E 3 restricts the z z z (212) has produced a non-negative majorant.
Bounding E 1 E_1 E 1 We now estimate the contribution to the double integral in (209) from the first range; we must show that the contribution is o ( ( log X ) 2 k − 1 ) o((\log X)^{2k-1}) o (( log X ) 2 k − 1 ) u ≪ h u\ll h u ≪ h (207) . Recall that
E 1 = [ 0 , X H Z ] = [ 0 , 1 h Z ] , Y 1 : = 1 h Z . E_1=\left[0,\frac{X}{HZ}\right]=\left[0,\frac{1}{hZ}\right],\qquad Y_1:=\frac{1}{hZ}. E 1 = [ 0 , H Z X ] = [ 0 , h Z 1 ] , Y 1 := h Z 1 . After taking absolute values and applying (212) , the E 1 E_1 E 1 ∣ z ∣ ≤ Y 1 |z|\le Y_1 ∣ z ∣ ≤ Y 1 ∣ z − s ∣ ≤ Y 1 |z-s|\le Y_1 ∣ z − s ∣ ≤ Y 1 z ↦ s − z z\mapsto s-z z ↦ s − z ∣ z ∣ ≤ h − 1 / 2 |z| \le h^{-1/2} ∣ z ∣ ≤ h − 1/2 (209) , which can only enlarge this non-negative upper bound, it is enough to estimate
1 h ∫ ∣ s ∣ ≤ S ∣ g ^ ( 1 + i s ) ∣ ∫ ∣ z ∣ ≤ h − 1 / 2 ∣ ζ ′ ζ ( σ ∗ + i z ) ∣ 2 k min ( u 2 , ∣ σ ∗ + i z ∣ − 2 ) d z d s . \frac{1}{h}\int_{|s|\le S}|\widehat{g}(1+is)|\int_{|z|\le h^{-1/2}}\left|\frac{\zeta'}{\zeta}(\sigma_*+iz)\right|^{2k}\min(u^2,|\sigma_*+iz|^{-2})\,dz\,ds. h 1 ∫ ∣ s ∣ ≤ S ∣ g ( 1 + i s ) ∣ ∫ ∣ z ∣ ≤ h − 1/2 ζ ζ ′ ( σ ∗ + i z ) 2 k min ( u 2 , ∣ σ ∗ + i z ∣ − 2 ) d z d s . On ∣ z ∣ < 1 |z|<1 ∣ z ∣ < 1 (217) . On 1 ≤ ∣ z ∣ ≤ h − 1 / 2 1\le|z|\le h^{-1/2} 1 ≤ ∣ z ∣ ≤ h − 1/2 (220) , the bound (221) , and the evenness of the integrand give
∫ 1 ≤ ∣ z ∣ ≤ h − 1 / 2 ∣ ζ ′ ζ ( σ ∗ + i z ) ∣ 2 k d z ≪ A ( h − 1 / 2 ) ≪ h − 1 / 2 ( log X ) 2 k . \int_{1\le|z|\le h^{-1/2}}\left|\frac{\zeta'}{\zeta}(\sigma_*+iz)\right|^{2k}dz\ll A(h^{-1/2})\ll h^{-1/2}(\log X)^{2k}. ∫ 1 ≤ ∣ z ∣ ≤ h − 1/2 ζ ζ ′ ( σ ∗ + i z ) 2 k d z ≪ A ( h − 1/2 ) ≪ h − 1/2 ( log X ) 2 k . Using min ( u 2 , ∣ σ ∗ + i z ∣ − 2 ) ≤ u 2 \min(u^2,|\sigma_*+iz|^{-2})\le u^2 min ( u 2 , ∣ σ ∗ + i z ∣ − 2 ) ≤ u 2 (215) , the preceding double integral is
≪ u 2 h ( 1 + A ( h − 1 / 2 ) ) ∫ R ∣ g ^ ( 1 + i s ) ∣ d s ≪ h 1 / 2 ( log X ) 2 k = o ( ( log X ) 2 k − 1 ) , \begin{aligned}
\ll\frac{u^2}{h}\left(1+A(h^{-1/2})\right)\int_{\mathbb{R}}|\widehat{g}(1+is)|\,ds \\
&\ll h^{1/2}(\log X)^{2k}=o\left((\log X)^{2k-1}\right),
\end{aligned} ≪ h u 2 ( 1 + A ( h − 1/2 ) ) ∫ R ∣ g ( 1 + i s ) ∣ d s ≪ h 1/2 ( log X ) 2 k = o ( ( log X ) 2 k − 1 ) , where u ≪ h u\ll h u ≪ h h 1 / 2 = X − 1 / ( 2 α ) h^{1/2}=X^{-1/(2\alpha)} h 1/2 = X − 1/ ( 2 α ) h − 1 / 2 ≤ ∣ z ∣ ≤ Y 1 h^{-1/2}\le|z|\le Y_1 h − 1/2 ≤ ∣ z ∣ ≤ Y 1 (212) would give only
O ( ( log X ) 2 k Z ) , O\left(\frac{(\log X)^{2k}}{Z}\right), O ( Z ( log X ) 2 k ) , which is not sufficient to obtain o ( ( log X ) 2 k − 1 ) o((\log X)^{2k-1}) o (( log X ) 2 k − 1 ) (209) , allowing (42) to provide the required decay in s s s s s s
A s ( Y ) : = ∫ 0 Y ( − ζ ′ ζ ( σ ∗ + i z ) ) k ( − ζ ′ ζ ( σ ∗ − i z + i s ) ) k d z . (224) A_s(Y):=\int_0^Y\left(-\frac{\zeta'}{\zeta}(\sigma_*+iz)\right)^k\left(-\frac{\zeta'}{\zeta}(\sigma_*-iz+is)\right)^k dz. \tag*{(224)} A s ( Y ) := ∫ 0 Y ( − ζ ζ ′ ( σ ∗ + i z ) ) k ( − ζ ζ ′ ( σ ∗ − i z + i s ) ) k d z . (224) The lower limit agrees with the averaging interval in (42) . We apply that estimate at averaging height Y Y Y z 1 = σ ∗ − 1 2 = 1 log T z_1=\sigma_*-\frac{1}{2}=\frac{1}{\log T} z 1 = σ ∗ − 2 1 = l o g T 1 z 2 = 1 2 − σ ∗ − i s = − 1 log T − i s z_2=\frac{1}{2}-\sigma_*-is=-\frac{1}{\log T}-is z 2 = 2 1 − σ ∗ − i s = − l o g T 1 − i s h − 1 / 2 ≤ Y ≤ Y 1 h^{-1/2}\le Y\le Y_1 h − 1/2 ≤ Y ≤ Y 1 log Y ≍ log X \log Y\asymp\log X log Y ≍ log X
∣ s ∣ ≤ ( log X ) η ≪ ( log Y ) η . |s|\le(\log X)^\eta\ll(\log Y)^\eta. ∣ s ∣ ≤ ( log X ) η ≪ ( log Y ) η . Thus the imaginary-shift conditions in (42) are satisfied; the real-part conditions follow from Y ≤ Y 1 ≤ T Y\le Y_1\le T Y ≤ Y 1 ≤ T
∣ z 1 − z 2 ∣ = ∣ 2 log T + i s ∣ ≥ ∣ s ∣ , ∣ z 1 − z 2 ∣ ≥ 2 log T . (225) |z_1-z_2|=\left|\frac{2}{\log T}+is\right|\ge|s|,\qquad|z_1-z_2|\ge\frac{2}{\log T}. \tag*{(225)} ∣ z 1 − z 2 ∣ = log T 2 + i s ≥ ∣ s ∣ , ∣ z 1 − z 2 ∣ ≥ log T 2 . (225) We will now use the above inequalities to bound the right-hand side of (42) . The second inequality in (225) gives
( log Y ) 2 k ( 1 + 1 ∣ z 1 − z 2 ∣ log Y ) 2 k = ( log Y + 1 ∣ z 1 − z 2 ∣ ) 2 k ≪ ( log X ) 2 k , (226) (\log Y)^{2k}\left(1+\frac{1}{|z_1-z_2|\log Y}\right)^{2k}=\left(\log Y+\frac{1}{|z_1-z_2|}\right)^{2k}\ll(\log X)^{2k}, \tag*{(226)} ( log Y ) 2 k ( 1 + ∣ z 1 − z 2 ∣ log Y 1 ) 2 k = ( log Y + ∣ z 1 − z 2 ∣ 1 ) 2 k ≪ ( log X ) 2 k , (226) while the first inequality implies min ( ∣ z 1 − z 2 ∣ , 1 ) ≥ min ( ∣ s ∣ , 1 ) \min(|z_1-z_2|,1)\ge\min(|s|,1) min ( ∣ z 1 − z 2 ∣ , 1 ) ≥ min ( ∣ s ∣ , 1 ) (42) satisfies
( 1 + min ( ∣ z 1 − z 2 ∣ , 1 ) log Y ) − 2 ≪ ( 1 + min ( ∣ s ∣ , 1 ) log ( 2 Y ) ) − 2 , \left(1+\min(|z_1-z_2|,1)\log Y\right)^{-2}\ll\left(1+\min(|s|,1)\log(2Y)\right)^{-2}, ( 1 + min ( ∣ z 1 − z 2 ∣ , 1 ) log Y ) − 2 ≪ ( 1 + min ( ∣ s ∣ , 1 ) log ( 2 Y ) ) − 2 , where we used log ( 2 Y ) ≍ log Y \log(2Y)\asymp\log Y log ( 2 Y ) ≍ log Y (226) , this gives
∣ A s ( Y ) ∣ ≪ Y ( log X ) 2 k D Y ( s ) − 2 , D Y ( s ) : = 1 + min ( 1 , ∣ s ∣ ) log ( 2 Y ) , (227) |A_s(Y)|\ll Y(\log X)^{2k}D_Y(s)^{-2},\qquad D_Y(s):=1+\min(1,|s|)\log(2Y), \tag*{(227)} ∣ A s ( Y ) ∣ ≪ Y ( log X ) 2 k D Y ( s ) − 2 , D Y ( s ) := 1 + min ( 1 , ∣ s ∣ ) log ( 2 Y ) , (227) uniformly for ∣ s ∣ ≤ ( log X ) η |s|\le(\log X)^\eta ∣ s ∣ ≤ ( log X ) η h − 1 / 2 ≤ Y ≤ Y 1 h^{-1/2}\le Y\le Y_1 h − 1/2 ≤ Y ≤ Y 1 A s A_s A s
F ( z , s , u ) ( σ ∗ + i z ) ( σ ∗ − i z + i s ) , \frac{F(z,s,u)}{(\sigma_*+iz)(\sigma_*-iz+is)}, ( σ ∗ + i z ) ( σ ∗ − i z + i s ) F ( z , s , u ) , where F F F (211) . We now bound this factor and its derivative with respect to z z z ( 1 + u ) w − 1 w = ∫ 0 u ( 1 + r ) w − 1 d r \frac{(1+u)^{w}-1}{w}=\int_0^u(1+r)^{w-1}\,dr w ( 1 + u ) w − 1 = ∫ 0 u ( 1 + r ) w − 1 d r
∣ F ( z , s , u ) ( σ ∗ + i z ) ( σ ∗ − i z + i s ) ∣ ≪ u 2 . (228) \left|\frac{F(z,s,u)}{(\sigma_*+iz)(\sigma_*-iz+is)}\right|\ll u^2. \tag*{(228)} ( σ ∗ + i z ) ( σ ∗ − i z + i s ) F ( z , s , u ) ≪ u 2 . (228) Differentiating under the integral sign gives one extra factor log ( 1 + r ) ≪ r \log(1+r)\ll r log ( 1 + r ) ≪ r
∣ ∂ z ( F ( z , s , u ) ( σ ∗ + i z ) ( σ ∗ − i z + i s ) ) ∣ ≪ u 3 . (229) \left|\partial_z\left(\frac{F(z,s,u)}{(\sigma_*+iz)(\sigma_*-iz+is)}\right)\right|\ll u^3. \tag*{(229)} ∂ z ( ( σ ∗ + i z ) ( σ ∗ − i z + i s ) F ( z , s , u ) ) ≪ u 3 . (229) Thus the short-interval kernel is slowly varying on the E 1 E_1 E 1 A s A_s A s (222) , gives
∫ h − 1 / 2 Y 1 ( − ζ ′ ζ ( σ ∗ + i z ) ) k ( − ζ ′ ζ ( σ ∗ − i z + i s ) ) k F ( z , s , u ) ( σ ∗ + i z ) ( σ ∗ − i z + i s ) d z = [ F ′ ( z , s , u ) ( σ ∗ + i z ) ( σ ∗ − i z + i s ) A s ( z ) ] h − 1 / 2 Y 1 − ∫ h − 1 / 2 Y 1 A s ( z ) ∂ z ( F ( z , s , u ) ( σ ∗ + i z ) ( σ ∗ − i z + i s ) ) d z . (230) \begin{aligned}
\int_{h^{-1/2}}^{Y_1}\left(-\frac{\zeta'}{\zeta}(\sigma_*+iz)\right)^k\left(-\frac{\zeta'}{\zeta}(\sigma_*-iz+is)\right)^k\frac{F(z,s,u)}{(\sigma_*+iz)(\sigma_*-iz+is)}\,dz
={}&\left[\frac{F'(z,s,u)}{(\sigma_*+iz)(\sigma_*-iz+is)}A_s(z)\right]_{h^{-1/2}}^{Y_1}\\
&-\int_{h^{-1/2}}^{Y_1}A_s(z)\partial_z\left(\frac{F(z,s,u)}{(\sigma_*+iz)(\sigma_*-iz+is)}\right)\,dz. \tag*{(230)}
\end{aligned} ∫ h − 1/2 Y 1 ( − ζ ζ ′ ( σ ∗ + i z ) ) k ( − ζ ζ ′ ( σ ∗ − i z + i s ) ) k ( σ ∗ + i z ) ( σ ∗ − i z + i s ) F ( z , s , u ) d z = [ ( σ ∗ + i z ) ( σ ∗ − i z + i s ) F ′ ( z , s , u ) A s ( z ) ] h − 1/2 Y 1 − ∫ h − 1/2 Y 1 A s ( z ) ∂ z ( ( σ ∗ + i z ) ( σ ∗ − i z + i s ) F ( z , s , u ) ) d z . (230) Using (227) , (228) , and (229) , the absolute value of this expression is
≪ u 2 ( log X ) 2 k ( Y 1 D Y 1 ( s ) − 2 + h − 1 / 2 D h − 1 / 2 ( s ) − 2 ) + u 3 ( log X ) 2 k ∫ h − 1 / 2 Y 1 Y D Y ( s ) − 2 d Y . \ll u^2(\log X)^{2k}\left(Y_1D_{Y_1}(s)^{-2}+h^{-1/2}D_{h^{-1/2}}(s)^{-2}\right)+u^3(\log X)^{2k}\int_{h^{-1/2}}^{Y_1}YD_Y(s)^{-2}\,dY. ≪ u 2 ( log X ) 2 k ( Y 1 D Y 1 ( s ) − 2 + h − 1/2 D h − 1/2 ( s ) − 2 ) + u 3 ( log X ) 2 k ∫ h − 1/2 Y 1 Y D Y ( s ) − 2 d Y . After multiplying by ∣ g ^ ( 1 + i s ) ∣ |\widehat{g}(1+is)| ∣ g ( 1 + i s ) ∣
∫ R ∣ g ^ ( 1 + i s ) ∣ D Y ( s ) − 2 d s ≪ 1 log ( 2 Y ) . (231) \int_{\mathbb{R}}|\widehat{g}(1+is)|D_Y(s)^{-2}\,ds\ll\frac{1}{\log(2Y)}. \tag*{(231)} ∫ R ∣ g ( 1 + i s ) ∣ D Y ( s ) − 2 d s ≪ log ( 2 Y ) 1 . (231) This follows from (214) by splitting the s s s ∣ s ∣ = 1 / log ( 2 Y ) |s|=1/\log(2Y) ∣ s ∣ = 1/ log ( 2 Y ) ∣ s ∣ = 1 |s|=1 ∣ s ∣ = 1 (231) to the endpoint and integral terms in (230) , multiplying by h − 1 ∣ g ^ ( 1 + i s ) ∣ h^{-1}|\widehat{g}(1+is)| h − 1 ∣ g ( 1 + i s ) ∣ ∣ s ∣ ≤ S |s|\le S ∣ s ∣ ≤ S (209) with h − 1 / 2 ≤ z ≤ Y 1 h^{-1/2}\le z\le Y_1 h − 1/2 ≤ z ≤ Y 1
≪ 1 h u 2 ( log X ) 2 k ( Y 1 log ( 2 Y 1 ) + h − 1 / 2 log ( 2 h − 1 / 2 ) ) + 1 h u 3 ( log X ) 2 k ∫ h − 1 / 2 Y 1 Y log ( 2 Y ) d Y . \ll\frac{1}{h}u^2(\log X)^{2k}\left(\frac{Y_1}{\log(2Y_1)}+\frac{h^{-1/2}}{\log(2h^{-1/2})}\right)+\frac{1}{h}u^3(\log X)^{2k}\int_{h^{-1/2}}^{Y_1}\frac{Y}{\log(2Y)}\,dY. ≪ h 1 u 2 ( log X ) 2 k ( log ( 2 Y 1 ) Y 1 + log ( 2 h − 1/2 ) h − 1/2 ) + h 1 u 3 ( log X ) 2 k ∫ h − 1/2 Y 1 log ( 2 Y ) Y d Y . Including the previously estimated range ∣ z ∣ ≤ h − 1 / 2 |z|\le h^{-1/2} ∣ z ∣ ≤ h − 1/2 0 ≤ z ≤ Y 1 0\le z\le Y_1 0 ≤ z ≤ Y 1 z z z E 1 E_1 E 1 (209) is
≪ 1 h u 2 ( log X ) 2 k ( Y 1 log ( 2 Y 1 ) + h − 1 / 2 log ( 2 h − 1 / 2 ) ) + 1 h u 3 ( log X ) 2 k ∫ h − 1 / 2 Y 1 Y log ( 2 Y ) d Y + o ( ( log X ) 2 k − 1 ) . \ll\frac{1}{h}u^2(\log X)^{2k}\left(\frac{Y_1}{\log(2Y_1)}+\frac{h^{-1/2}}{\log(2h^{-1/2})}\right)+\frac{1}{h}u^3(\log X)^{2k}\int_{h^{-1/2}}^{Y_1}\frac{Y}{\log(2Y)}\,dY+o\left((\log X)^{2k-1}\right). ≪ h 1 u 2 ( log X ) 2 k ( log ( 2 Y 1 ) Y 1 + log ( 2 h − 1/2 ) h − 1/2 ) + h 1 u 3 ( log X ) 2 k ∫ h − 1/2 Y 1 log ( 2 Y ) Y d Y + o ( ( log X ) 2 k − 1 ) . Using
∫ h − 1 / 2 Y 1 Y log ( 2 Y ) d Y ≪ Y 1 2 log ( 2 Y 1 ) , \int_{h^{-1/2}}^{Y_1}\frac{Y}{\log(2Y)}\,dY\ll\frac{Y_1^2}{\log(2Y_1)}, ∫ h − 1/2 Y 1 log ( 2 Y ) Y d Y ≪ log ( 2 Y 1 ) Y 1 2 , together with u ≪ h u\ll h u ≪ h Y 1 = 1 / ( h Z ) Y_1=1/(hZ) Y 1 = 1/ ( h Z ) log ( 2 Y 1 ) ≍ log ( 2 h − 1 / 2 ) ≍ log X \log(2Y_1)\asymp\log(2h^{-1/2})\asymp\log X log ( 2 Y 1 ) ≍ log ( 2 h − 1/2 ) ≍ log X
≪ ( log X ) 2 k − 1 Z + h 1 / 2 ( log X ) 2 k − 1 + ( log X ) 2 k − 1 Z 2 + o ( ( log X ) 2 k − 1 ) = o ( ( log X ) 2 k − 1 ) , \ll\frac{(\log X)^{2k-1}}{Z}+h^{1/2}(\log X)^{2k-1}+\frac{(\log X)^{2k-1}}{Z^2}+o\left((\log X)^{2k-1}\right)=o\left((\log X)^{2k-1}\right), ≪ Z ( log X ) 2 k − 1 + h 1/2 ( log X ) 2 k − 1 + Z 2 ( log X ) 2 k − 1 + o ( ( log X ) 2 k − 1 ) = o ( ( log X ) 2 k − 1 ) , because Z → ∞ Z\to\infty Z → ∞ h → 0 h\to0 h → 0
Therefore, the positive-z z z E 1 E_1 E 1 − Y 1 ≤ z ≤ − h − 1 / 2 -Y_1\le z\le-h^{-1/2} − Y 1 ≤ z ≤ − h − 1/2 ∣ s ∣ |s| ∣ s ∣ g ^ \widehat{g} g
Bounding E 2 E_2 E 2 E 1 E_1 E 1 ∣ s ∣ ≤ ( log X ) η |s| \le(\log X)^\eta ∣ s ∣ ≤ ( log X ) η E 2 : = [ Z h , Z h log T ] E_2 := \left[\frac{Z}{h}, \frac{Z}{h}\log T\right] E 2 := [ h Z , h Z log T ] ∣ σ ∗ + i z ∣ ≍ ∣ σ ∗ − i z + i s ∣ ≍ ∣ z ∣ |\sigma_*+iz| \asymp|\sigma_* - iz+is| \asymp|z| ∣ σ ∗ + i z ∣ ≍ ∣ σ ∗ − i z + i s ∣ ≍ ∣ z ∣ (209) from E 2 E_2 E 2 s s s
s 0 : = ( log log X ) 2 log X . s_0 := \frac{(\log\log X)^2}{\log X}. s 0 := log X ( log log X ) 2 . First suppose ∣ s ∣ < s 0 |s|<s_0 ∣ s ∣ < s 0 (212) and repeating the arguments made in the derivation of (213) , with the E 2 E_2 E 2 ∣ z ∣ |z| ∣ z ∣ ∣ z − s ∣ |z-s| ∣ z − s ∣
≪ 1 h ∫ ∣ s ∣ < s 0 ∣ g ^ ( 1 + i s ) ∣ ∫ Z / h Z log T / h ∣ ζ ′ ζ ( σ ∗ + i z ) ∣ 2 k z − 2 d z d s . \ll\frac{1}{h}\int_{|s|<s_0}|\widehat{g}(1+is)|\int_{Z/h}^{Z\log T/h}\left|\frac{\zeta'}{\zeta}(\sigma_*+iz)\right|^{2k}z^{-2}\,dz\,ds. ≪ h 1 ∫ ∣ s ∣ < s 0 ∣ g ( 1 + i s ) ∣ ∫ Z / h Z l o g T / h ζ ζ ′ ( σ ∗ + i z ) 2 k z − 2 d z d s . By the same Stieltjes argument as in the calculation following (222) , together with (221) , we have
∫ Z / h Z log T / h ∣ ζ ′ ζ ( σ ∗ + i z ) ∣ 2 k z − 2 d z = ∫ Z / h Z log T / h z − 2 d A ( z ) ≪ ( log X ) 2 k ( h Z + ∫ Z / h Z log T / h z − 2 d z ) ≪ h Z ( log X ) 2 k . (232) \begin{aligned}
\int_{Z/h}^{Z\log T/h}\left|\frac{\zeta'}{\zeta}(\sigma_*+iz)\right|^{2k}z^{-2}\,dz
=
\int_{Z/h}^{Z\log T/h}z^{-2}\,dA(z) \\
&\ll(\log X)^{2k}\left(\frac{h}{Z}+\int_{Z/h}^{Z\log T/h}z^{-2}\,dz\right) \\
&\ll\frac{h}{Z}(\log X)^{2k}. \tag*{(232)}
\end{aligned} ∫ Z / h Z l o g T / h ζ ζ ′ ( σ ∗ + i z ) 2 k z − 2 d z = ∫ Z / h Z l o g T / h z − 2 d A ( z ) ≪ ( log X ) 2 k ( Z h + ∫ Z / h Z l o g T / h z − 2 d z ) ≪ Z h ( log X ) 2 k . (232) Therefore
≪ 1 h ( log log X ) 2 log X h Z ( log X ) 2 k = ( log X ) 2 k − 1 ( log log X ) 2 Z = o ( ( log X ) 2 k − 1 ) , \ll\frac{1}{h}\frac{(\log\log X)^2}{\log X}\frac{h}{Z}(\log X)^{2k}
= \frac{(\log X)^{2k-1}(\log\log X)^2}{Z}
= o((\log X)^{2k-1}), ≪ h 1 log X ( log log X ) 2 Z h ( log X ) 2 k = Z ( log X ) 2 k − 1 ( log log X ) 2 = o (( log X ) 2 k − 1 ) , since Z / ( log log X ) 2 → ∞ Z/(\log\log X)^2\to\infty Z / ( log log X ) 2 → ∞ s 0 ≤ ∣ s ∣ ≤ ( log X ) η s_0\le|s|\le(\log X)^\eta s 0 ≤ ∣ s ∣ ≤ ( log X ) η (212) would lose the required saving in s s s (43) for this range and then use partial summation, as in the treatment of E 1 E_1 E 1 E 2 E_2 E 2 F ( z , s , u ) = O ( 1 ) F(z,s,u)=O(1) F ( z , s , u ) = O ( 1 )
∣ F ( z , s , u ) ( σ ∗ + i z ) ( σ ∗ − i z + i s ) ∣ ≪ ∣ z ∣ − 2 . \left|\frac{F(z,s,u)}{(\sigma_*+iz)(\sigma_*-iz+is)}\right|\ll|z|^{-2}. ( σ ∗ + i z ) ( σ ∗ − i z + i s ) F ( z , s , u ) ≪ ∣ z ∣ − 2 . Furthermore,
∂ z F ( z , s , u ) = i log ( 1 + u ) ( ( 1 + u ) σ ∗ − i z + i s − ( 1 + u ) σ ∗ + i z ) , \partial_zF(z,s,u)=i\log(1+u)\left((1+u)^{\sigma_*-iz+is}-(1+u)^{\sigma_*+iz}\right), ∂ z F ( z , s , u ) = i log ( 1 + u ) ( ( 1 + u ) σ ∗ − i z + i s − ( 1 + u ) σ ∗ + i z ) , and because u ≪ h u\ll h u ≪ h ∂ z F ( z , s , u ) ≪ h \partial_zF(z,s,u)\ll h ∂ z F ( z , s , u ) ≪ h
∂ z ( F ( z , s , u ) ( σ ∗ + i z ) ( σ ∗ − i z + i s ) ) ≪ h ∣ z ∣ − 2 + ∣ z ∣ − 3 . \partial_z\left(\frac{F(z,s,u)}{(\sigma_*+iz)(\sigma_*-iz+is)}\right)\ll h|z|^{-2}+|z|^{-3}. ∂ z ( ( σ ∗ + i z ) ( σ ∗ − i z + i s ) F ( z , s , u ) ) ≪ h ∣ z ∣ − 2 + ∣ z ∣ − 3 . For fixed s s s A s ( Y ) A_s(Y) A s ( Y ) (224) . Then we apply the proposition and, uniformly on Z h ≤ Y ≤ Z h log T \frac{Z}{h}\le Y\le\frac{Z}{h}\log T h Z ≤ Y ≤ h Z log T
A s ( Y ) ≪ Y ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 . (233) A_s(Y)\ll Y(\log X)^{2k}(1+\min(1,|s|)\log X)^{-2}. \tag*{(233)} A s ( Y ) ≪ Y ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 . (233) as log Y ≍ log X \log Y\asymp\log X log Y ≍ log X E 2 E_2 E 2 (42) , the simplified estimate (43) omits the factor
( 1 + 1 ∣ z 1 − z 2 ∣ log Y ) 2 k , \left(1+\frac{1}{|z_1-z_2|\log Y}\right)^{2k}, ( 1 + ∣ z 1 − z 2 ∣ log Y 1 ) 2 k , and therefore requires ∣ z 1 − z 2 ∣ log Y |z_1-z_2|\log Y ∣ z 1 − z 2 ∣ log Y
∣ z 1 − z 2 ∣ log Y ≥ ∣ s ∣ log Y ≥ s 0 log Y ≍ ( log log X ) 2 . |z_1-z_2|\log Y\ge|s|\log Y\ge s_0\log Y\asymp(\log\log X)^2. ∣ z 1 − z 2 ∣ log Y ≥ ∣ s ∣ log Y ≥ s 0 log Y ≍ ( log log X ) 2 . Thus this additional condition is satisfied, and (43) gives (233) uniformly over the stated ranges of s s s Y Y Y
∫ Z h Z h log T ( − ζ ′ ζ ( σ ∗ + i z ) ) k ( − ζ ′ ζ ( σ ∗ − i z + i s ) ) k F ( z , s , u ) ( σ ∗ + i z ) ( σ ∗ − i z + i s ) d z = [ F ( z , s , u ) ( σ ∗ + i z ) ( σ ∗ − i z + i s ) A s ( z ) ] Z h Z h log T − ∫ Z h Z h log T A s ( z ) ∂ z ( F ( z , s , u ) ( σ ∗ + i z ) ( σ ∗ − i z + i s ) ) d z . \begin{aligned}
\int_{\frac{Z}{h}}^{\frac{Z}{h}\log T}\left(-\frac{\zeta'}{\zeta}(\sigma_*+iz)\right)^k\left(-\frac{\zeta'}{\zeta}(\sigma_*-iz+is)\right)^k\frac{F(z,s,u)}{(\sigma_*+iz)(\sigma_*-iz+is)}\,dz \\
&=\left[\frac{F(z,s,u)}{(\sigma_*+iz)(\sigma_*-iz+is)}A_s(z)\right]_{\frac{Z}{h}}^{\frac{Z}{h}\log T}-\int_{\frac{Z}{h}}^{\frac{Z}{h}\log T}A_s(z)\partial_z\left(\frac{F(z,s,u)}{(\sigma_*+iz)(\sigma_*-iz+is)}\right)\,dz.
\end{aligned} ∫ h Z h Z l o g T ( − ζ ζ ′ ( σ ∗ + i z ) ) k ( − ζ ζ ′ ( σ ∗ − i z + i s ) ) k ( σ ∗ + i z ) ( σ ∗ − i z + i s ) F ( z , s , u ) d z = [ ( σ ∗ + i z ) ( σ ∗ − i z + i s ) F ( z , s , u ) A s ( z ) ] h Z h Z l o g T − ∫ h Z h Z l o g T A s ( z ) ∂ z ( ( σ ∗ + i z ) ( σ ∗ − i z + i s ) F ( z , s , u ) ) d z . Using the bounds for F ( z , s , u ) F(z,s,u) F ( z , s , u )
≪ ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 h Z . \ll(\log X)^{2k}(1+\min(1,|s|)\log X)^{-2}\frac{h}{Z}. ≪ ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 Z h . while the integral term contributes
≪ ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 ∫ Z h Z h log T z ( h z − 2 + z − 3 ) d z ≪ ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 ( h log log X + H X Z ) . \begin{aligned}
\ll(\log X)^{2k}(1+\min(1,|s|)\log X)^{-2}\int_{\frac{Z}{h}}^{\frac{Z}{h}\log T}z(hz^{-2}+z^{-3})\,dz \\
&\ll(\log X)^{2k}(1+\min(1,|s|)\log X)^{-2}\left(h\log\log X+\frac{H}{XZ}\right).
\end{aligned} ≪ ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 ∫ h Z h Z l o g T z ( h z − 2 + z − 3 ) d z ≪ ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 ( h log log X + X Z H ) . We then have
∫ Z h Z h log T ( − ζ ′ ζ ( σ ∗ + i z ) ) k ( − ζ ′ ζ ( σ ∗ − i z + i s ) ) k F ( z , s , u ) ( σ ∗ + i z ) ( σ ∗ − i z + i s ) d z ≪ ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 ( h Z + h log log X ) . \begin{aligned}
\int_{\frac{Z}{h}}^{\frac{Z}{h}\log T}\left(-\frac{\zeta'}{\zeta}(\sigma_*+iz)\right)^k\left(-\frac{\zeta'}{\zeta}(\sigma_*-iz+is)\right)^k\frac{F(z,s,u)}{(\sigma_*+iz)(\sigma_*-iz+is)}\,dz \\
&\ll(\log X)^{2k}(1+\min(1,|s|)\log X)^{-2}\left(\frac{h}{Z}+h\log\log X\right).
\end{aligned} ∫ h Z h Z l o g T ( − ζ ζ ′ ( σ ∗ + i z ) ) k ( − ζ ζ ′ ( σ ∗ − i z + i s ) ) k ( σ ∗ + i z ) ( σ ∗ − i z + i s ) F ( z , s , u ) d z ≪ ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 ( Z h + h log log X ) . Multiplying by ∣ g ^ ( 1 + i s ) ∣ |\widehat{g}(1+is)| ∣ g ( 1 + i s ) ∣ ( log log X ) 2 log X ≤ ∣ s ∣ ≤ ( log X ) η \frac{(\log\log X)^2}{\log X}\leq|s|\leq(\log X)^\eta l o g X ( l o g l o g X ) 2 ≤ ∣ s ∣ ≤ ( log X ) η 1 / h 1/h 1/ h
≪ 1 h ( log X ) 2 k ( h Z + h log log X ) ∫ ∣ s ∣ ≥ ( log log X ) 2 / log X ∣ g ^ ( 1 + i s ) ∣ ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 d s . (234) \ll\frac{1}{h}(\log X)^{2k}\left(\frac{h}{Z}+h\log\log X\right)
\int_{|s|\geq(\log\log X)^2/\log X}|\widehat{g}(1+is)|(1+\min(1,|s|)\log X)^{-2}\,ds. \tag*{(234)} ≪ h 1 ( log X ) 2 k ( Z h + h log log X ) ∫ ∣ s ∣ ≥ ( l o g l o g X ) 2 / l o g X ∣ g ( 1 + i s ) ∣ ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 d s . (234) To bound the above, we split the s s s s 0 ≤ ∣ s ∣ ≤ 1 s_0\leq|s|\leq1 s 0 ≤ ∣ s ∣ ≤ 1 ∣ s ∣ > 1 |s|>1 ∣ s ∣ > 1 g ^ \widehat{g} g
∫ ∣ s ∣ ≥ s 0 ∣ g ^ ( 1 + i s ) ∣ ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 d s ≪ ∫ s 0 1 ( 1 + s log X ) − 2 d s + ( log X ) − 2 . \int_{|s|\geq s_0}|\widehat{g}(1+is)|(1+\min(1,|s|)\log X)^{-2}\,ds\ll\int_{s_0}^{1}(1+s\log X)^{-2}\,ds+(\log X)^{-2}. ∫ ∣ s ∣ ≥ s 0 ∣ g ( 1 + i s ) ∣ ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 d s ≪ ∫ s 0 1 ( 1 + s log X ) − 2 d s + ( log X ) − 2 . Changing variables v = s log X v=s\log X v = s log X
∫ s 0 1 ( 1 + s log X ) − 2 d s = 1 log X ∫ ( log log X ) 2 log X ( 1 + v ) − 2 d v ≪ 1 log X ( log log X ) 2 . (235) \int_{s_0}^{1}(1+s\log X)^{-2}\,ds=\frac{1}{\log X}\int_{(\log\log X)^2}^{\log X}(1+v)^{-2}\,dv\ll\frac{1}{\log X(\log\log X)^2}. \tag*{(235)} ∫ s 0 1 ( 1 + s log X ) − 2 d s = log X 1 ∫ ( l o g l o g X ) 2 l o g X ( 1 + v ) − 2 d v ≪ log X ( log log X ) 2 1 . (235) From this, we can bound (234) and arrive at
≪ ( log X ) 2 k ( 1 Z + log log X ) 1 log X ( log log X ) 2 ≪ ( log X ) 2 k − 1 log log X + ( log X ) 2 k − 1 Z ( log log X ) 2 = o ( ( log X ) 2 k − 1 ) . \begin{aligned}
\ll(\log X)^{2k}\left(\frac{1}{Z}+\log\log X\right)\frac{1}{\log X(\log\log X)^2} \\
&\ll\frac{(\log X)^{2k-1}}{\log\log X}+\frac{(\log X)^{2k-1}}{Z(\log\log X)^2}=o\left((\log X)^{2k-1}\right).
\end{aligned} ≪ ( log X ) 2 k ( Z 1 + log log X ) log X ( log log X ) 2 1 ≪ log log X ( log X ) 2 k − 1 + Z ( log log X ) 2 ( log X ) 2 k − 1 = o ( ( log X ) 2 k − 1 ) . The two ranges of s s s z z z E 2 E_2 E 2 z z z E 2 E_2 E 2 o ( ( log X ) 2 k − 1 ) o((\log X)^{2k-1}) o (( log X ) 2 k − 1 )
Bounding E 3 E_3 E 3 E 3 = [ Z h log T , T ] E_3 = \left[\frac{Z}{h}\log T,T\right] E 3 = [ h Z log T , T ] ∣ s ∣ ≤ ( log X ) η |s| \le(\log X)^\eta ∣ s ∣ ≤ ( log X ) η ∣ z ∣ ≥ Z h log T |z| \ge\frac{Z}{h}\log T ∣ z ∣ ≥ h Z log T E 2 E_2 E 2 ∣ σ ∗ + i z ∣ ≍ ∣ σ ∗ + i ( s − z ) ∣ ≍ ∣ z ∣ |\sigma_*+iz| \asymp|\sigma_*+i(s-z)| \asymp|z| ∣ σ ∗ + i z ∣ ≍ ∣ σ ∗ + i ( s − z ) ∣ ≍ ∣ z ∣ s s s E 3 E_3 E 3 (212) and proceed as we did for the near-diagonal part of E 2 E_2 E 2 y = s − z y=s-z y = s − z E 3 E_3 E 3 (208) is bounded by
≪ 1 h ∫ ∣ s ∣ ≤ ( log X ) η ∣ g ^ ( 1 + i s ) ∣ ∫ Z h log T T ∣ ζ ′ ζ ( σ ∗ + i z ) ∣ 2 k z − 2 d z d s . \ll\frac{1}{h}\int_{|s|\le(\log X)^\eta}|\widehat{g}(1+is)|\int_{\frac{Z}{h}\log T}^{T}\left|\frac{\zeta'}{\zeta}(\sigma_*+iz)\right|^{2k}z^{-2}\,dz\,ds. ≪ h 1 ∫ ∣ s ∣ ≤ ( l o g X ) η ∣ g ( 1 + i s ) ∣ ∫ h Z l o g T T ζ ζ ′ ( σ ∗ + i z ) 2 k z − 2 d z d s . By (220) , followed by Stieltjes integration by parts using (221) , we have
∫ Z h log T T ∣ ζ ′ ζ ( σ ∗ + i z ) ∣ 2 k z − 2 d z = ∫ Z h log T T z − 2 d A ( z ) ≪ h ( log X ) 2 k Z log T . \int_{\frac{Z}{h}\log T}^{T}\left|\frac{\zeta'}{\zeta}(\sigma_*+iz)\right|^{2k}z^{-2}\,dz=\int_{\frac{Z}{h}\log T}^{T}z^{-2}\,dA(z)\ll\frac{h(\log X)^{2k}}{Z\log T}. ∫ h Z l o g T T ζ ζ ′ ( σ ∗ + i z ) 2 k z − 2 d z = ∫ h Z l o g T T z − 2 d A ( z ) ≪ Z log T h ( log X ) 2 k . Moreover, (215) gives
∫ ∣ s ∣ ≤ ( log X ) η ∣ g ^ ( 1 + i s ) ∣ d s ≪ g 1. \int_{|s|\le(\log X)^\eta}|\widehat{g}(1+is)|\,ds\ll_g 1. ∫ ∣ s ∣ ≤ ( l o g X ) η ∣ g ( 1 + i s ) ∣ d s ≪ g 1. This means that the region E 3 E_3 E 3
≪ 1 h h ( log X ) 2 k Z log T ∫ ∣ s ∣ ≤ ( log X ) η ∣ g ^ ( 1 + i s ) ∣ d s ≪ ( log X ) 2 k Z log T . \ll\frac{1}{h}\frac{h(\log X)^{2k}}{Z\log T}\int_{|s|\le(\log X)^\eta}|\widehat{g}(1+is)|\,ds\ll\frac{(\log X)^{2k}}{Z\log T}. ≪ h 1 Z log T h ( log X ) 2 k ∫ ∣ s ∣ ≤ ( l o g X ) η ∣ g ( 1 + i s ) ∣ d s ≪ Z log T ( log X ) 2 k . As log T ≍ log X \log T\asymp\log X log T ≍ log X
≪ ( log X ) 2 k − 1 Z = o ( ( log X ) 2 k − 1 ) . \ll\frac{(\log X)^{2k-1}}{Z}=o\left((\log X)^{2k-1}\right). ≪ Z ( log X ) 2 k − 1 = o ( ( log X ) 2 k − 1 ) . The negative z z z Z = ( log log X ) A Z=(\log\log X)^A Z = ( log log X ) A A > 2 A>2 A > 2 E 1 E_1 E 1 E 3 E_3 E 3 1 / Z 1/Z 1/ Z E 2 E_2 E 2 Z − 1 ( log log X ) 2 = o ( 1 ) Z^{-1}(\log\log X)^2=o(1) Z − 1 ( log log X ) 2 = o ( 1 ) E 2 E_2 E 2 1 / log log X 1/\log\log X 1/ log log X (208) and the total contribution of z ∈ [ 0 , 1 h Z ] ∪ [ Z h , T ] z\in[0,\frac{1}{hZ}]\cup[\frac{Z}{h},T] z ∈ [ 0 , h Z 1 ] ∪ [ h Z , T ] (206) is o ( ( log X ) 2 k − 1 ) o((\log X)^{2k-1}) o (( log X ) 2 k − 1 )
The central range
We now focus on the central contribution given by
1 H X ∫ R g ( x X ) ∣ 1 2 π ∫ 1 h Z ≤ ∣ t ∣ ≤ Z h ( − ζ ′ ζ ( σ ∗ + i t ) ) k ( x + H ) σ ∗ + i t − x σ ∗ + i t σ ∗ + i t d t ∣ 2 d x . (236) \frac{1}{HX}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)\left|\frac{1}{2\pi}\int_{\frac{1}{hZ}\le|t|\le\frac{Z}{h}}\left(-\frac{\zeta'}{\zeta}(\sigma_*+it)\right)^k\frac{(x+H)^{\sigma_*+it}-x^{\sigma_*+it}}{\sigma_*+it}\,dt\right|^2dx. \tag*{(236)} H X 1 ∫ R g ( X x ) 2 π 1 ∫ h Z 1 ≤ ∣ t ∣ ≤ h Z ( − ζ ζ ′ ( σ ∗ + i t ) ) k σ ∗ + i t ( x + H ) σ ∗ + i t − x σ ∗ + i t d t 2 d x . (236) Expanding the squared modulus and putting s = t 1 − t 2 s=t_1-t_2 s = t 1 − t 2 z = t 1 z=t_1 z = t 1
1 H ( 2 π ) 2 ∬ 1 h Z ≤ ∣ z ∣ , ∣ z − s ∣ ≤ Z h ( − ζ ′ ζ ( σ ∗ + i z ) ) k ( − ζ ′ ζ ( σ ∗ − i z + i s ) ) k L ( z , s ) d z d s , (237) \frac{1}{H(2\pi)^2}\iint_{\frac{1}{hZ}\le|z|,|z-s|\le\frac{Z}{h}}\left(-\frac{\zeta'}{\zeta}(\sigma_*+iz)\right)^k\left(-\frac{\zeta'}{\zeta}(\sigma_*-iz+is)\right)^k\mathcal{L}(z,s)\,dz\,ds, \tag*{(237)} H ( 2 π ) 2 1 ∬ h Z 1 ≤ ∣ z ∣ , ∣ z − s ∣ ≤ h Z ( − ζ ζ ′ ( σ ∗ + i z ) ) k ( − ζ ζ ′ ( σ ∗ − i z + i s ) ) k L ( z , s ) d z d s , (237) where
L ( z , s ) : = 1 X ∫ R g ( x X ) ( x + H ) σ ∗ + i z − x σ ∗ + i z σ ∗ + i z ( x + H ) σ ∗ − i z + i s − x σ ∗ − i z + i s σ ∗ − i z + i s d x = 1 X ∫ R g ( x X ) F ( z , s , H / x ) ( σ ∗ + i z ) ( σ ∗ − i z + i s ) x 2 σ ∗ + i s d x . (238) \begin{aligned}
\mathcal{L}(z,s):=\frac{1}{X}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)\frac{(x+H)^{\sigma_*+iz}-x^{\sigma_*+iz}}{\sigma_*+iz}\frac{(x+H)^{\sigma_*-iz+is}-x^{\sigma_*-iz+is}}{\sigma_*-iz+is}\,dx \\
&=\frac{1}{X}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)\frac{F(z,s,H/x)}{(\sigma_*+iz)(\sigma_*-iz+is)}x^{2\sigma_*+is}\,dx. \tag*{(238)}
\end{aligned} L ( z , s ) := X 1 ∫ R g ( X x ) σ ∗ + i z ( x + H ) σ ∗ + i z − x σ ∗ + i z σ ∗ − i z + i s ( x + H ) σ ∗ − i z + i s − x σ ∗ − i z + i s d x = X 1 ∫ R g ( X x ) ( σ ∗ + i z ) ( σ ∗ − i z + i s ) F ( z , s , H / x ) x 2 σ ∗ + i s d x . (238) Here F F F (211) . To bound L \mathcal{L} L g ( x / X ) g(x/X) g ( x / X ) H / x ≍ h H/x\asymp h H / x ≍ h u = H / x u=H/x u = H / x x ∂ x = − u ∂ u x\partial_x=-u\partial_u x ∂ x = − u ∂ u u ∂ u u\partial_u u ∂ u ( 1 + u ) a (1+u)^a ( 1 + u ) a ( 1 + u ) b (1+u)^b ( 1 + u ) b O ( u ∣ a ∣ ) O(u|a|) O ( u ∣ a ∣ ) O ( u ∣ b ∣ ) O(u|b|) O ( u ∣ b ∣ ) m ≥ 0 m\ge0 m ≥ 0
∣ ( x ∂ x ) m F ( z , s , H / x ) ∣ ≪ m ( 1 + h ( 1 + ∣ z ∣ + ∣ z − s ∣ ) ) m . (239) \left|(x\partial_x)^mF(z,s,H/x)\right|\ll_m\left(1+h(1+|z|+|z-s|)\right)^m. \tag*{(239)} ∣ ( x ∂ x ) m F ( z , s , H / x ) ∣ ≪ m ( 1 + h ( 1 + ∣ z ∣ + ∣ z − s ∣ ) ) m . (239) Moreover, differentiating F F F z z z log ( 1 + u ) ≪ u ≪ h \log(1+u)\ll u\ll h log ( 1 + u ) ≪ u ≪ h
∣ ( x ∂ x ) m ∂ z F ( z , s , H / x ) ∣ ≪ m h ( 1 + h ( 1 + ∣ z ∣ + ∣ z − s ∣ ) ) m . (240) \left|(x\partial_x)^m\partial_zF(z,s,H/x)\right|\ll_m h\left(1+h(1+|z|+|z-s|)\right)^m. \tag*{(240)} ∣ ( x ∂ x ) m ∂ z F ( z , s , H / x ) ∣ ≪ m h ( 1 + h ( 1 + ∣ z ∣ + ∣ z − s ∣ ) ) m . (240) We integrate by parts ℓ ≥ 1 \ell\ge1 ℓ ≥ 1 (238) using x ∂ x x 2 σ ∗ + i s = ( 2 σ ∗ + i s ) x 2 σ ∗ + i s x\partial_x x^{2\sigma_*+is}=(2\sigma_*+is)x^{2\sigma_*+is} x ∂ x x 2 σ ∗ + i s = ( 2 σ ∗ + i s ) x 2 σ ∗ + i s ( 1 + ∣ s ∣ ) − 1 (1+|s|)^{-1} ( 1 + ∣ s ∣ ) − 1 x ∂ x x\partial_x x ∂ x ( x ∂ x ) j (x\partial_x)^j ( x ∂ x ) j j ≤ ℓ j\le\ell j ≤ ℓ g ( x / X ) g(x/X) g ( x / X ) (239) and the smooth compact support of g ( x / X ) g(x/X) g ( x / X )
L ( z , s ) ≪ ℓ X ( 1 + h ( 1 + ∣ z ∣ + ∣ z − s ∣ ) ) ℓ ( 1 + ∣ s ∣ ) ℓ ∣ ( σ ∗ + i z ) ( σ ∗ − i z + i s ) ∣ . \mathcal{L}(z,s) \ll_{\ell} \frac{X\left(1+h(1+|z|+|z-s|)\right)^{\ell}}{(1+|s|)^{\ell}\left|(\sigma_*+iz)(\sigma_*-iz+is)\right|}. L ( z , s ) ≪ ℓ ( 1 + ∣ s ∣ ) ℓ ∣ ( σ ∗ + i z ) ( σ ∗ − i z + i s ) ∣ X ( 1 + h ( 1 + ∣ z ∣ + ∣ z − s ∣ ) ) ℓ . Here the factor X X X x x x X − 1 ∫ g ( x / X ) x 2 σ ∗ d x ≪ X 2 σ ∗ ≍ X X^{-1}\int g(x/X)x^{2\sigma_*}\,dx\ll X^{2\sigma_*}\asymp X X − 1 ∫ g ( x / X ) x 2 σ ∗ d x ≪ X 2 σ ∗ ≍ X z z z
∂ z L ( z , s ) ≪ ℓ X ( 1 + h ( 1 + ∣ z ∣ + ∣ z − s ∣ ) ) ℓ ( 1 + ∣ s ∣ ) ℓ ∣ ( σ ∗ + i z ) ( σ ∗ − i z + i s ) ∣ ( h + 1 1 + ∣ z ∣ + 1 1 + ∣ z − s ∣ ) . \partial_z\mathcal{L}(z,s) \ll_{\ell} \frac{X\left(1+h(1+|z|+|z-s|)\right)^{\ell}}{(1+|s|)^{\ell}\left|(\sigma_*+iz)(\sigma_*-iz+is)\right|}\left(h+\frac{1}{1+|z|}+\frac{1}{1+|z-s|}\right). ∂ z L ( z , s ) ≪ ℓ ( 1 + ∣ s ∣ ) ℓ ∣ ( σ ∗ + i z ) ( σ ∗ − i z + i s ) ∣ X ( 1 + h ( 1 + ∣ z ∣ + ∣ z − s ∣ ) ) ℓ ( h + 1 + ∣ z ∣ 1 + 1 + ∣ z − s ∣ 1 ) . These estimates hold for every integer ℓ ≥ 0 \ell\ge0 ℓ ≥ 0 ℓ = 0 \ell=0 ℓ = 0 (238) . In the central range, we may assume 1 + h ( 1 + ∣ z ∣ + ∣ z − s ∣ ) ≪ Z 1+h(1+|z|+|z-s|)\ll Z 1 + h ( 1 + ∣ z ∣ + ∣ z − s ∣ ) ≪ Z h + 1 1 + ∣ z ∣ + 1 1 + ∣ z − s ∣ ≪ h Z h+\frac{1}{1+|z|}+\frac{1}{1+|z-s|}\ll hZ h + 1 + ∣ z ∣ 1 + 1 + ∣ z − s ∣ 1 ≪ h Z
L ( z , s ) ≪ ℓ Z ℓ X ( 1 + ∣ s ∣ ) ℓ ∣ ( σ ∗ + i z ) ( σ ∗ − i z + i s ) ∣ . (241) \mathcal{L}(z,s)\ll_{\ell}\frac{Z^\ell X}{(1+|s|)^\ell\left|(\sigma_*+iz)(\sigma_*-iz+is)\right|}. \tag*{(241)} L ( z , s ) ≪ ℓ ( 1 + ∣ s ∣ ) ℓ ∣ ( σ ∗ + i z ) ( σ ∗ − i z + i s ) ∣ Z ℓ X . (241) and
∂ z L ( z , s ) ≪ ℓ Z ℓ + 1 H ( 1 + ∣ s ∣ ) ℓ ∣ ( σ ∗ + i z ) ( σ ∗ − i z + i s ) ∣ . (242) \partial_z\mathcal{L}(z,s)\ll_{\ell}\frac{Z^{\ell+1}H}{(1+|s|)^\ell\left|(\sigma_*+iz)(\sigma_*-iz+is)\right|}. \tag*{(242)} ∂ z L ( z , s ) ≪ ℓ ( 1 + ∣ s ∣ ) ℓ ∣ ( σ ∗ + i z ) ( σ ∗ − i z + i s ) ∣ Z ℓ + 1 H . (242) We first estimate the portion of (237) for which ∣ s ∣ > ( log X ) η |s|>(\log X)^\eta ∣ s ∣ > ( log X ) η (241) and the same decoupling used in (212) , we have
≪ ℓ Z ℓ X H ( 1 + ∣ s ∣ ) ℓ ( ∣ ζ ′ ζ ( σ ∗ + i z ) ∣ 2 k ∣ σ ∗ + i z ∣ − 2 + ∣ ζ ′ ζ ( σ ∗ − i z + i s ) ∣ 2 k ∣ σ ∗ − i z + i s ∣ − 2 ) . \ll_{\ell}\frac{Z^\ell X}{H(1+|s|)^\ell}\left(\left|\frac{\zeta'}{\zeta}(\sigma_*+iz)\right|^{2k}|\sigma_*+iz|^{-2}+\left|\frac{\zeta'}{\zeta}(\sigma_*-iz+is)\right|^{2k}|\sigma_*-iz+is|^{-2}\right). ≪ ℓ H ( 1 + ∣ s ∣ ) ℓ Z ℓ X ( ζ ζ ′ ( σ ∗ + i z ) 2 k ∣ σ ∗ + i z ∣ − 2 + ζ ζ ′ ( σ ∗ − i z + i s ) 2 k ∣ σ ∗ − i z + i s ∣ − 2 ) . The second term is treated by a variable change and enlarging the resulting domain as in the large-∣ s ∣ |s| ∣ s ∣ z z z s s s z z z (237) is
≪ ℓ Z ℓ X H ( 1 + ∣ s ∣ ) ℓ ∫ 1 / ( h Z ) Z / h ∣ ζ ′ ζ ( σ ∗ + i t ) ∣ 2 k t − 2 d t . \ll_{\ell}\frac{Z^\ell X}{H(1+|s|)^\ell}\int_{1/(hZ)}^{Z/h}\left|\frac{\zeta'}{\zeta}(\sigma_*+it)\right|^{2k}t^{-2}\,dt. ≪ ℓ H ( 1 + ∣ s ∣ ) ℓ Z ℓ X ∫ 1/ ( h Z ) Z / h ζ ζ ′ ( σ ∗ + i t ) 2 k t − 2 d t . By a partial summation similar to (232) , we get
∫ 1 / ( h Z ) Z / h ∣ ζ ′ ζ ( σ ∗ + i t ) ∣ 2 k t − 2 d t = ∫ 1 / ( h Z ) Z / h t − 2 d A ( t ) ≪ h Z ( log X ) 2 k , (243) \int_{1/(hZ)}^{Z/h}\left|\frac{\zeta'}{\zeta}(\sigma_*+it)\right|^{2k}t^{-2}\,dt=\int_{1/(hZ)}^{Z/h}t^{-2}\,dA(t)\ll hZ(\log X)^{2k}, \tag*{(243)} ∫ 1/ ( h Z ) Z / h ζ ζ ′ ( σ ∗ + i t ) 2 k t − 2 d t = ∫ 1/ ( h Z ) Z / h t − 2 d A ( t ) ≪ h Z ( log X ) 2 k , (243) which follows from (221) . Therefore the contribution of ∣ s ∣ > ( log X ) η |s|>(\log X)^\eta ∣ s ∣ > ( log X ) η
≪ ℓ Z ℓ + 1 ( log X ) 2 k ∫ ∣ s ∣ > ( log X ) η ( 1 + ∣ s ∣ ) − ℓ d s . \ll_{\ell}Z^{\ell+1}(\log X)^{2k}\int_{|s|>(\log X)^\eta}(1+|s|)^{-\ell}\,ds. ≪ ℓ Z ℓ + 1 ( log X ) 2 k ∫ ∣ s ∣ > ( l o g X ) η ( 1 + ∣ s ∣ ) − ℓ d s . Choosing ℓ \ell ℓ η ( ℓ − 1 ) > 1 \eta(\ell-1)>1 η ( ℓ − 1 ) > 1 o ( ( log X ) 2 k − 1 ) o((\log X)^{2k-1}) o (( log X ) 2 k − 1 )
Next, we analyse the range
( log X ) − 1 + ε < ∣ s ∣ ≤ ( log X ) η , (\log X)^{-1+\varepsilon}<|s|\le(\log X)^\eta, ( log X ) − 1 + ε < ∣ s ∣ ≤ ( log X ) η , where ε \varepsilon ε 0 < ε < 1 0<\varepsilon<1 0 < ε < 1 ∣ s ∣ ≪ ∣ z ∣ |s|\ll|z| ∣ s ∣ ≪ ∣ z ∣ z z z ( z , s ) ↦ ( − z , − s ) (z,s)\mapsto(-z,-s) ( z , s ) ↦ ( − z , − s ) ℓ = 0 \ell=0 ℓ = 0
L ( z , s ) ≪ X h z 2 , ∂ z L ( z , s ) ≪ X ( h z − 2 + z − 3 ) . (244) \mathcal{L}(z,s)\ll\frac{X}{hz^2},\qquad\partial_z\mathcal{L}(z,s)\ll X(hz^{-2}+z^{-3}). \tag*{(244)} L ( z , s ) ≪ h z 2 X , ∂ z L ( z , s ) ≪ X ( h z − 2 + z − 3 ) . (244) The argument establishing (227) also applies throughout the central range. For
1 h Z ≤ Y ≤ Z h , \frac{1}{hZ}\le Y\le\frac{Z}{h}, h Z 1 ≤ Y ≤ h Z , we have log Y ≍ log X \log Y \asymp\log X log Y ≍ log X Y ≤ Z / h = o ( T ) Y \le Z/h = o(T) Y ≤ Z / h = o ( T ) (42) continue to hold. In addition, as ∣ s ∣ ≤ ( log X ) η ≪ ( log Y ) η |s| \le(\log X)^\eta\ll(\log Y)^\eta ∣ s ∣ ≤ ( log X ) η ≪ ( log Y ) η
∣ A s ( Y ) ∣ ≪ Y ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 , (245) |A_s(Y)| \ll Y(\log X)^{2k}\left(1+\min(1,|s|)\log X\right)^{-2}, \tag*{(245)} ∣ A s ( Y ) ∣ ≪ Y ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 , (245) uniformly for ∣ s ∣ ≤ ( log X ) η |s| \le(\log X)^\eta ∣ s ∣ ≤ ( log X ) η 1 h Z ≤ Y ≤ Z h \frac{1}{hZ} \le Y \le\frac{Z}{h} h Z 1 ≤ Y ≤ h Z
As in the treatment of E 2 E_2 E 2 ( log log X ) 2 ( log X ) − 1 ≤ ∣ s ∣ ≤ ( log X ) η (\log\log X)^2(\log X)^{-1} \le|s| \le(\log X)^\eta ( log log X ) 2 ( log X ) − 1 ≤ ∣ s ∣ ≤ ( log X ) η (212) would lose the required decay in s s s A s A_s A s (230) . Following the convention introduced before bounding E 1 E_1 E 1 ∣ z − s ∣ |z-s| ∣ z − s ∣ (237) ; partial summation is understood to be applied on the actual sub-interval, uniformly in its endpoints. Using (244) and (245) , we obtain
∫ 1 h Z Z h ( − ζ ′ ζ ( σ ∗ + i z ) ) k ( − ζ ′ ζ ( σ ∗ − i z + i s ) ) k L ( z , s ) d z ≪ H Z ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 , \int_{\frac{1}{hZ}}^{\frac{Z}{h}}\left(-\frac{\zeta'}{\zeta}(\sigma_*+iz)\right)^k\left(-\frac{\zeta'}{\zeta}(\sigma_*-iz+is)\right)^k\mathcal{L}(z,s)\,dz
\ll HZ(\log X)^{2k}\left(1+\min(1,|s|)\log X\right)^{-2}, ∫ h Z 1 h Z ( − ζ ζ ′ ( σ ∗ + i z ) ) k ( − ζ ζ ′ ( σ ∗ − i z + i s ) ) k L ( z , s ) d z ≪ H Z ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 , uniformly for s s s 1 / H 1/H 1/ H s s s
≪ Z ( log X ) 2 k ∫ ( log X ) − 1 + ε < ∣ s ∣ ≤ ( log X ) η ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 d s ≪ Z ( log X ) 2 k ( ( log X ) − 1 − ε + ( log X ) − 2 + η ) . \begin{aligned}
\ll Z(\log X)^{2k}\int_{(\log X)^{-1+\varepsilon}<|s|\le(\log X)^\eta}\left(1+\min(1,|s|)\log X\right)^{-2}\,ds \\
&\ll Z(\log X)^{2k}\left((\log X)^{-1-\varepsilon}+(\log X)^{-2+\eta}\right).
\end{aligned} ≪ Z ( log X ) 2 k ∫ ( l o g X ) − 1 + ε < ∣ s ∣ ≤ ( l o g X ) η ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 d s ≪ Z ( log X ) 2 k ( ( log X ) − 1 − ε + ( log X ) − 2 + η ) . As we chose Z = ( log log X ) A Z=(\log\log X)^A Z = ( log log X ) A η < 1 \eta<1 η < 1 o ( ( log X ) 2 k − 1 ) o((\log X)^{2k-1}) o (( log X ) 2 k − 1 ) ∣ s ∣ ≤ ( log X ) − 1 + ε |s|\le(\log X)^{-1+\varepsilon} ∣ s ∣ ≤ ( log X ) − 1 + ε κ = σ ∗ − 1 2 = 1 log T \kappa=\sigma_*-\frac{1}{2}=\frac{1}{\log T} κ = σ ∗ − 2 1 = l o g T 1 z 1 = κ z_1=\kappa z 1 = κ z 2 = − κ − i s z_2=-\kappa-is z 2 = − κ − i s z 1 − z 2 = 2 κ + i s z_1-z_2=2\kappa+is z 1 − z 2 = 2 κ + i s 0 < ξ < 1 0<\xi<1 0 < ξ < 1 k ≥ 3 k\ge3 k ≥ 3 0 < ξ < c 0<\xi<c 0 < ξ < c k = 1 k=1 k = 1 k = 2 k=2 k = 2
1 h Z ≤ Y ≤ Z h , \frac{1}{hZ}\le Y\le\frac{Z}{h}, h Z 1 ≤ Y ≤ h Z , we apply Conjecture 1 with its averaging height T T T Y Y Y Y Y Y Y \mathcal{Y} Y Y = 1 \mathcal{Y}=1 Y = 1 X X X
∣ ℑ z 2 ∣ = ∣ ℑ ( z 1 − z 2 ) ∣ = ∣ s ∣ ≤ ( log X ) − 1 + ε < 1. |\Im z_2|=|\Im(z_1-z_2)|=|s|\le(\log X)^{-1+\varepsilon}<1. ∣ℑ z 2 ∣ = ∣ℑ ( z 1 − z 2 ) ∣ = ∣ s ∣ ≤ ( log X ) − 1 + ε < 1. Furthermore, Y ≤ Z / h = o ( T ) Y\le Z/h=o(T) Y ≤ Z / h = o ( T )
0 < − ℜ z 2 = 1 log T ≤ 3 log Y , 0 < ℜ ( z 1 − z 2 ) = 2 log T ≤ 3 log Y , 0<-\Re z_2=\frac{1}{\log T}\le\frac{3}{\log Y},\qquad0<\Re(z_1-z_2)=\frac{2}{\log T}\le\frac{3}{\log Y}, 0 < − ℜ z 2 = log T 1 ≤ log Y 3 , 0 < ℜ ( z 1 − z 2 ) = log T 2 ≤ log Y 3 , so the real-part conditions in Conjecture 1 are also satisfied. For the cumulative moment A s ( Y ) A_s(Y) A s ( Y ) (224) , the conjecture gives
A s ( Y ) = Y k ! 2 ( 2 π i ) 2 k ∫ C u k ∫ C v k Y 1 2 ( ∑ j = 1 k u j − ∑ ℓ = 1 k v ℓ ) ∏ j , ℓ = 1 k ( v ℓ − u j − 2 κ − i s ) × ∏ 1 ≤ i < j ≤ k ( u i − u j ) 2 ∏ 1 ≤ i < j ≤ k ( v i − v j ) 2 ( ∑ j = 1 k 1 u j + ∑ ℓ = 1 k 1 v ℓ − 2 κ − i s − log Y 2 ) k × ( ∑ j = 1 k 1 u j + 2 κ + i s + ∑ ℓ = 1 k 1 v ℓ + log Y 2 ) k ( 2 κ + i s ) − k 2 ∏ j = 1 k u j − k d u j ∏ ℓ = 1 k v ℓ − k d v ℓ + O ( Y ( log Y ) 2 k − 1 + ξ ( 1 + min ( 1 , ∣ s ∣ ) log Y ) − θ k , c ) . (246) \begin{aligned}
A_s(Y)={}&\frac{Y}{k!^2(2\pi i)^{2k}}\int_{\mathcal{C}_u^k}\int_{\mathcal{C}_v^k}Y^{\frac{1}{2}\left(\sum_{j=1}^k u_j-\sum_{\ell=1}^k v_\ell\right)}
\prod_{j,\ell=1}^k(v_\ell-u_j-2\kappa-is)\\
&\quad\times\prod_{1\le i<j\le k}(u_i-u_j)^2\prod_{1\le i<j\le k}(v_i-v_j)^2
\left(\sum_{j=1}^k\frac{1}{u_j}+\sum_{\ell=1}^k\frac{1}{v_\ell-2\kappa-is}-\frac{\log Y}{2}\right)^k\\
&\quad\times\left(\sum_{j=1}^k\frac{1}{u_j+2\kappa+is}+\sum_{\ell=1}^k\frac{1}{v_\ell}+\frac{\log Y}{2}\right)^k
(2\kappa+is)^{-k^2}\prod_{j=1}^k u_j^{-k}\,du_j\prod_{\ell=1}^k v_\ell^{-k}\,dv_\ell\\
&\quad+O\left(Y(\log Y)^{2k-1+\xi}\left(1+\min(1,|s|)\log Y\right)^{-\theta_{k,c}}\right). \tag*{(246)}
\end{aligned} A s ( Y ) = k ! 2 ( 2 π i ) 2 k Y ∫ C u k ∫ C v k Y 2 1 ( ∑ j = 1 k u j − ∑ ℓ = 1 k v ℓ ) j , ℓ = 1 ∏ k ( v ℓ − u j − 2 κ − i s ) × 1 ≤ i < j ≤ k ∏ ( u i − u j ) 2 1 ≤ i < j ≤ k ∏ ( v i − v j ) 2 ( j = 1 ∑ k u j 1 + ℓ = 1 ∑ k v ℓ − 2 κ − i s 1 − 2 log Y ) k × ( j = 1 ∑ k u j + 2 κ + i s 1 + ℓ = 1 ∑ k v ℓ 1 + 2 log Y ) k ( 2 κ + i s ) − k 2 j = 1 ∏ k u j − k d u j ℓ = 1 ∏ k v ℓ − k d v ℓ + O ( Y ( log Y ) 2 k − 1 + ξ ( 1 + min ( 1 , ∣ s ∣ ) log Y ) − θ k , c ) . (246) We now compare L ( z , s ) \mathcal{L}(z,s) L ( z , s ) L ( z , 0 ) \mathcal{L}(z,0) L ( z , 0 )
Q ( w , u ) : = ( 1 + u ) w − 1 w . Q(w,u) := \frac{(1+u)^w-1}{w}. Q ( w , u ) := w ( 1 + u ) w − 1 . After setting x = X v x=Xv x = X v (238) , we have
X − i s L ( z , s ) = X 2 σ ∗ ∫ R g ( v ) v 2 σ ∗ + i s Q ( σ ∗ + i z , h v ) Q ( σ ∗ − i z + i s , h v ) d v . (247) X^{-is}\mathcal{L}(z,s)=X^{2\sigma_*}\int_{\mathbb{R}}g(v)v^{2\sigma_*+is}Q\left(\sigma_*+iz,\frac{h}{v}\right)Q\left(\sigma_*-iz+is,\frac{h}{v}\right)\,dv. \tag*{(247)} X − i s L ( z , s ) = X 2 σ ∗ ∫ R g ( v ) v 2 σ ∗ + i s Q ( σ ∗ + i z , v h ) Q ( σ ∗ − i z + i s , v h ) d v . (247) This form isolates the factor X i s X^{is} X i s x i s = X i s v i s x^{is}=X^{is}v^{is} x i s = X i s v i s g g g v ≍ 1 v\asymp1 v ≍ 1 v i s v^{is} v i s O g ( ∣ s ∣ ) O_g(|s|) O g ( ∣ s ∣ ) s = 0 s=0 s = 0 Q Q Q s s s h / v ≍ h h/v\asymp h h / v ≍ h g g g ∣ s ∣ ≤ ( log X ) − 1 + ε |s|\leq(\log X)^{-1+\varepsilon} ∣ s ∣ ≤ ( log X ) − 1 + ε ∣ σ ∗ + i z ∣ ≍ ∣ σ ∗ − i z + i s ∣ ≍ ∣ z ∣ |\sigma_*+iz|\asymp|\sigma_*-iz+is|\asymp|z| ∣ σ ∗ + i z ∣ ≍ ∣ σ ∗ − i z + i s ∣ ≍ ∣ z ∣ log ( 1 + u ) ≪ h \log(1+u)\ll h log ( 1 + u ) ≪ h u ≍ h u\asymp h u ≍ h Q ( w , u ) Q(w,u) Q ( w , u )
Q ( w , u ) ≪ ∣ z ∣ − 1 , ∂ w Q ( w , u ) ≪ h ∣ z ∣ − 1 + ∣ z ∣ − 2 , ∂ w 2 Q ( w , u ) ≪ h 2 ∣ z ∣ − 1 + h ∣ z ∣ − 2 + ∣ z ∣ − 3 , (248) \begin{aligned}
Q(w,u)&\ll|z|^{-1},\\
\partial_wQ(w,u)&\ll h|z|^{-1}+|z|^{-2},\\
\partial_w^2Q(w,u)&\ll h^2|z|^{-1}+h|z|^{-2}+|z|^{-3}, \tag*{(248)}
\end{aligned} Q ( w , u ) ∂ w Q ( w , u ) ∂ w 2 Q ( w , u ) ≪ ∣ z ∣ − 1 , ≪ h ∣ z ∣ − 1 + ∣ z ∣ − 2 , ≪ h 2 ∣ z ∣ − 1 + h ∣ z ∣ − 2 + ∣ z ∣ − 3 , (248) whenever ℜ w = σ ∗ \Re w=\sigma_* ℜ w = σ ∗ ∣ w ∣ ≍ ∣ z ∣ |w|\asymp|z| ∣ w ∣ ≍ ∣ z ∣ u ≍ h u\asymp h u ≍ h (247) with respect to s s s
∂ s [ v i s Q ( σ ∗ − i z + i s , h v ) ] = i v i s [ ( log v ) Q ( σ ∗ − i z + i s , h v ) + ∂ w Q ( σ ∗ − i z + i s , h v ) ] . \partial_s\left[v^{is}Q\left(\sigma_*-iz+is,\frac{h}{v}\right)\right]
=iv^{is}\left[(\log v)Q\left(\sigma_*-iz+is,\frac{h}{v}\right)+\partial_wQ\left(\sigma_*-iz+is,\frac{h}{v}\right)\right]. ∂ s [ v i s Q ( σ ∗ − i z + i s , v h ) ] = i v i s [ ( log v ) Q ( σ ∗ − i z + i s , v h ) + ∂ w Q ( σ ∗ − i z + i s , v h ) ] . Using log v = O g ( 1 ) \log v=O_g(1) log v = O g ( 1 )
X 2 σ ∗ ∫ R ∣ g ( v ) ∣ v 2 σ ∗ d v ≪ g X , X^{2\sigma_*}\int_{\mathbb{R}}|g(v)|v^{2\sigma_*}\,dv\ll_g X, X 2 σ ∗ ∫ R ∣ g ( v ) ∣ v 2 σ ∗ d v ≪ g X , we obtain, uniformly in the small-s s s
∂ s ( X − i s L ( z , s ) ) ≪ X z 2 . (249) \partial_s\left(X^{-is}\mathcal{L}(z,s)\right)\ll\frac{X}{z^2}. \tag*{(249)} ∂ s ( X − i s L ( z , s ) ) ≪ z 2 X . (249) Differentiating this expression once with respect to z z z ∂ w Q \partial_wQ ∂ w Q ∂ w 2 Q \partial_w^2Q ∂ w 2 Q (248) similarly gives
∂ z ∂ s ( X − i s L ( z , s ) ) ≪ X ( h z − 2 + z − 3 ) . (250) \partial_z\partial_s\left(X^{-is}\mathcal{L}(z,s)\right)\ll X\left(hz^{-2}+z^{-3}\right). \tag*{(250)} ∂ z ∂ s ( X − i s L ( z , s ) ) ≪ X ( h z − 2 + z − 3 ) . (250) Integrating these estimates with respect to s s s 0 0 0 s s s
L ( z , s ) = X i s L ( z , 0 ) + O ( ∣ s ∣ X z 2 ) , (251) \mathcal{L}(z,s)=X^{is}\mathcal{L}(z,0)+O\left(|s|\frac{X}{z^2}\right), \tag*{(251)} L ( z , s ) = X i s L ( z , 0 ) + O ( ∣ s ∣ z 2 X ) , (251) and
∣ ∂ z ( L ( z , s ) − X i s L ( z , 0 ) ) ∣ ≪ ∣ s ∣ X ( h z − 2 + z − 3 ) . (252) \left|\partial_z\left(\mathcal{L}(z,s)-X^{is}\mathcal{L}(z,0)\right)\right|\ll|s|X\left(hz^{-2}+z^{-3}\right). \tag*{(252)} ∂ z ( L ( z , s ) − X i s L ( z , 0 ) ) ≪ ∣ s ∣ X ( h z − 2 + z − 3 ) . (252) To estimate the error introduced in (237) by replacing L ( z , s ) \mathcal{L}(z,s) L ( z , s ) X i s L ( z , 0 ) X^{is}\mathcal{L}(z,0) X i s L ( z , 0 ) A s A_s A s (224) , using the bound (245) . Write
R ( z , s ) : = L ( z , s ) − X i s L ( z , 0 ) . R(z,s):=\mathcal{L}(z,s)-X^{is}\mathcal{L}(z,0). R ( z , s ) := L ( z , s ) − X i s L ( z , 0 ) . The shifted logarithmic derivatives need not be estimated pointwise: partial summation transfers the z z z R ( z , s ) R(z,s) R ( z , s ) (251) and (252) are available. Applying partial summation in the form of (230) , together with (245) , gives
∣ ∫ 1 / ( h Z ) Z / h ( − ζ ′ ζ ( σ ∗ + i z ) ) k ( − ζ ′ ζ ( σ ∗ − i z + i s ) ) k R ( z , s ) d z ∣ ≪ ∣ s ∣ X ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 ( h Z + h Z + h ∫ 1 / ( h Z ) Z / h d z z + h ∫ 1 / ( h Z ) Z / h d z z 2 ) ≪ ∣ s ∣ H Z ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 . \begin{aligned}
\left|\int_{1/(hZ)}^{Z/h}\left(-\frac{\zeta'}{\zeta}(\sigma_*+iz)\right)^k\left(-\frac{\zeta'}{\zeta}(\sigma_*-iz+is)\right)^k R(z,s)\,dz\right| \\
\ll|s|X(\log X)^{2k}\left(1+\min(1,|s|)\log X\right)^{-2}
\left(hZ+\frac{h}{Z}+h\int_{1/(hZ)}^{Z/h}\frac{dz}{z}+h\int_{1/(hZ)}^{Z/h}\frac{dz}{z^2}\right) \\
\ll|s|HZ(\log X)^{2k}\left(1+\min(1,|s|)\log X\right)^{-2}.
\end{aligned} ∫ 1/ ( h Z ) Z / h ( − ζ ζ ′ ( σ ∗ + i z ) ) k ( − ζ ζ ′ ( σ ∗ − i z + i s ) ) k R ( z , s ) d z ≪ ∣ s ∣ X ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 ( h Z + Z h + h ∫ 1/ ( h Z ) Z / h z d z + h ∫ 1/ ( h Z ) Z / h z 2 d z ) ≪ ∣ s ∣ H Z ( log X ) 2 k ( 1 + min ( 1 , ∣ s ∣ ) log X ) − 2 . Here the first two terms in parentheses arise from the two endpoints in partial summation, while the final two arise from integrating the two terms in (252) . The negative z z z
After including the outside factor 1 / H 1/H 1/ H s s s L ( z , s ) \mathcal{L}(z,s) L ( z , s ) X i s L ( z , 0 ) X^{is}\mathcal{L}(z,0) X i s L ( z , 0 )
≪ Z ( log X ) 2 k ∫ ∣ s ∣ ≤ ( log X ) − 1 + ε ∣ s ∣ ( 1 + ∣ s ∣ log X ) − 2 d s = 2 Z ( log X ) 2 k ( log X ) 2 ∫ 0 ( log X ) ε v ( 1 + v ) 2 d v ≪ Z ( log X ) 2 k log log X ( log X ) 2 = o ( ( log X ) 2 k − 1 ) . \begin{aligned}
&\ll Z(\log X)^{2k}\int_{|s|\leq(\log X)^{-1+\varepsilon}} |s|(1+|s|\log X)^{-2}\,ds \\
&=\frac{2Z(\log X)^{2k}}{(\log X)^2}\int_0^{(\log X)^\varepsilon}\frac{v}{(1+v)^2}\,dv \\
&\ll Z(\log X)^{2k}\frac{\log\log X}{(\log X)^2}
=o\left((\log X)^{2k-1}\right).
\end{aligned} ≪ Z ( log X ) 2 k ∫ ∣ s ∣ ≤ ( l o g X ) − 1 + ε ∣ s ∣ ( 1 + ∣ s ∣ log X ) − 2 d s = ( log X ) 2 2 Z ( log X ) 2 k ∫ 0 ( l o g X ) ε ( 1 + v ) 2 v d v ≪ Z ( log X ) 2 k ( log X ) 2 log log X = o ( ( log X ) 2 k − 1 ) . As ∣ s ∣ ≤ ( log X ) − 1 + ε |s|\leq(\log X)^{-1+\varepsilon} ∣ s ∣ ≤ ( log X ) − 1 + ε 1 h Z ≤ ∣ z − s ∣ ≤ Z h \frac{1}{hZ}\leq|z-s|\leq\frac{Z}{h} h Z 1 ≤ ∣ z − s ∣ ≤ h Z 1 h Z ≤ ∣ z ∣ ≤ Z h \frac{1}{hZ}\leq|z|\leq\frac{Z}{h} h Z 1 ≤ ∣ z ∣ ≤ h Z O ( ∣ s ∣ ) O(|s|) O ( ∣ s ∣ ) (147) , applied with η = 1 \eta=1 η = 1
∣ ζ ′ ζ ( σ ∗ + i z ) ∣ + ∣ ζ ′ ζ ( σ ∗ − i z + i s ) ∣ ≪ ( log T ) 2 . \left|\frac{\zeta'}{\zeta}(\sigma_*+iz)\right|+\left|\frac{\zeta'}{\zeta}(\sigma_*-iz+is)\right|\ll(\log T)^2. ζ ζ ′ ( σ ∗ + i z ) + ζ ζ ′ ( σ ∗ − i z + i s ) ≪ ( log T ) 2 . In addition, (241) , with ℓ = 0 \ell=0 ℓ = 0 s = 0 s=0 s = 0
L ( z , 0 ) ≪ X ∣ σ ∗ + i z ∣ ∣ σ ∗ − i z ∣ ≪ X z 2 . \mathcal{L}(z,0)\ll\frac{X}{|\sigma_*+iz|\,|\sigma_*-iz|}\ll\frac{X}{z^2}. L ( z , 0 ) ≪ ∣ σ ∗ + i z ∣ ∣ σ ∗ − i z ∣ X ≪ z 2 X . As the boundary intervals have total length O ( ∣ s ∣ ) O(|s|) O ( ∣ s ∣ ) (147) and (241) , applied with ℓ = 0 \ell=0 ℓ = 0 s = 0 s=0 s = 0 (206) is
O ( h Z 2 ( log X ) 4 k − 2 + 2 ε ) = o ( ( log X ) 2 k − 1 ) . O\left(hZ^2(\log X)^{4k-2+2\varepsilon}\right)
=o\left((\log X)^{2k-1}\right). O ( h Z 2 ( log X ) 4 k − 2 + 2 ε ) = o ( ( log X ) 2 k − 1 ) . Thus the condition on ∣ z − s ∣ |z-s| ∣ z − s ∣ ∣ z ∣ |z| ∣ z ∣
1 H ( 2 π ) 2 ∬ 1 h Z ≤ ∣ z ∣ , ∣ z − s ∣ ≤ Z h ( − ζ ′ ζ ( σ ∗ + i z ) ) k ( − ζ ′ ζ ( σ ∗ − i z + i s ) ) k L ( z , s ) d z d s = 1 H ( 2 π ) 2 ∫ ∣ s ∣ ≤ ( log X ) − 1 + ε X i s ∫ 1 h Z ≤ ∣ z ∣ ≤ Z h L ( z , 0 ) ( − ζ ′ ζ ( σ ∗ + i z ) ) k ( − ζ ′ ζ ( σ ∗ − i z + i s ) ) k d z d s + o ( ( log X ) 2 k − 1 ) . \begin{aligned}
\frac{1}{H(2\pi)^2}\iint_{\frac{1}{hZ}\leq|z|,|z-s|\leq\frac{Z}{h}}
\left(-\frac{\zeta'}{\zeta}(\sigma_*+iz)\right)^k
\left(-\frac{\zeta'}{\zeta}(\sigma_*-iz+is)\right)^k
\mathcal{L}(z,s)\,dz\,ds \\
=\frac{1}{H(2\pi)^2}\int_{|s|\leq(\log X)^{-1+\varepsilon}}X^{is}
\int_{\frac{1}{hZ}\leq|z|\leq\frac{Z}{h}}
\mathcal{L}(z,0)
\left(-\frac{\zeta'}{\zeta}(\sigma_*+iz)\right)^k
\left(-\frac{\zeta'}{\zeta}(\sigma_*-iz+is)\right)^k\,dz\,ds
+o\left((\log X)^{2k-1}\right).
\end{aligned} H ( 2 π ) 2 1 ∬ h Z 1 ≤ ∣ z ∣ , ∣ z − s ∣ ≤ h Z ( − ζ ζ ′ ( σ ∗ + i z ) ) k ( − ζ ζ ′ ( σ ∗ − i z + i s ) ) k L ( z , s ) d z d s = H ( 2 π ) 2 1 ∫ ∣ s ∣ ≤ ( l o g X ) − 1 + ε X i s ∫ h Z 1 ≤ ∣ z ∣ ≤ h Z L ( z , 0 ) ( − ζ ζ ′ ( σ ∗ + i z ) ) k ( − ζ ζ ′ ( σ ∗ − i z + i s ) ) k d z d s + o ( ( log X ) 2 k − 1 ) . We insert (246) through partial summation in the z z z L ( z , 0 ) \mathcal{L}(z,0) L ( z , 0 ) (244) , so the conjectural error term contributes at most
≪ Z ( log X ) 2 k − 1 + ξ ∫ ∣ s ∣ ≤ ( log X ) − 1 + ε ( 1 + min ( 1 , ∣ s ∣ ) log X ) − θ k , c d s , \ll Z(\log X)^{2k-1+\xi}\int_{|s|\leq(\log X)^{-1+\varepsilon}}
\left(1+\min(1,|s|)\log X\right)^{-\theta_{k,c}}\,ds, ≪ Z ( log X ) 2 k − 1 + ξ ∫ ∣ s ∣ ≤ ( l o g X ) − 1 + ε ( 1 + min ( 1 , ∣ s ∣ ) log X ) − θ k , c d s , and, using the definition of θ k , c \theta_{k,c} θ k , c (125) , we have uniformly in all cases
∫ ∣ s ∣ ≤ ( log X ) − 1 + ε ( 1 + ∣ s ∣ log X ) − θ k , c d s ≪ 1 log X . \int_{|s|\leq(\log X)^{-1+\varepsilon}} (1+|s|\log X)^{-\theta_{k,c}}\,ds \ll\frac{1}{\log X}. ∫ ∣ s ∣ ≤ ( l o g X ) − 1 + ε ( 1 + ∣ s ∣ log X ) − θ k , c d s ≪ log X 1 . Thus the contribution of the conjectural error term is o ( ( log X ) 2 k − 1 ) o((\log X)^{2k-1}) o (( log X ) 2 k − 1 ) 0 < ξ < 1 0<\xi<1 0 < ξ < 1 k ≥ 3 k\geq3 k ≥ 3 0 < ξ < c 0<\xi<c 0 < ξ < c (246) . By (153) , this term is
Y k ! 2 ( 2 κ + i s ) − k 2 W k ( 2 κ + i s ; Y ) . \frac{Y}{k!^{2}}(2\kappa+is)^{-k^{2}}\mathcal{W}_{k}(2\kappa+is;Y). k ! 2 Y ( 2 κ + i s ) − k 2 W k ( 2 κ + i s ; Y ) . The two-centre decomposition leading to (162) , together with (164) –(169) , gives
∣ ∂ ∂ log Y ‾ ( ( 2 κ + i s ) − k 2 W k ( 2 κ + i s ; Y ) ) ∣ ≪ ( log X ) 2 k − 1 , (253) \left|\frac{\partial}{\partial\log\overline{Y}}\left((2\kappa+is)^{-k^{2}}\mathcal{W}_{k}(2\kappa+is;Y)\right)\right|\ll(\log X)^{2k-1}, \tag*{(253)} ∂ log Y ∂ ( ( 2 κ + i s ) − k 2 W k ( 2 κ + i s ; Y ) ) ≪ ( log X ) 2 k − 1 , (253) uniformly for
1 h Z ≤ Y ≤ Z h , ∣ s ∣ ≤ ( log X ) − 1 + ε . \frac{1}{hZ}\leq Y\leq\frac{Z}{h},\qquad|s|\leq(\log X)^{-1+\varepsilon}. h Z 1 ≤ Y ≤ h Z , ∣ s ∣ ≤ ( log X ) − 1 + ε . As ∣ log ( h Y ) ∣ ≤ log Z |\log(hY)|\leq\log Z ∣ log ( hY ) ∣ ≤ log Z (253) gives
∣ ( 2 κ + i s ) − k 2 W k ( 2 κ + i s ; Y ) − ( 2 κ + i s ) − k 2 W k ( 2 κ + i s ; 1 / h ) ∣ ≪ ( log Z ) ( log X ) 2 k − 1 . (254) \left|(2\kappa+is)^{-k^{2}}\mathcal{W}_{k}(2\kappa+is;Y)-(2\kappa+is)^{-k^{2}}\mathcal{W}_{k}(2\kappa+is;1/h)\right|
\ll(\log Z)(\log X)^{2k-1}. \tag*{(254)} ( 2 κ + i s ) − k 2 W k ( 2 κ + i s ; Y ) − ( 2 κ + i s ) − k 2 W k ( 2 κ + i s ; 1/ h ) ≪ ( log Z ) ( log X ) 2 k − 1 . (254) We now evaluate the remaining z z z L ( z , 0 ) \mathcal{L}(z,0) L ( z , 0 ) x x x
K x ( z ) : = ( x + H ) σ ∗ + i z − x σ ∗ + i z σ ∗ + i z = ∫ x x + H y σ ∗ + i z − 1 d y . K_x(z):=\frac{(x+H)^{\sigma_*+iz}-x^{\sigma_*+iz}}{\sigma_*+iz}=\int_x^{x+H}y^{\sigma_*+iz-1}\,dy. K x ( z ) := σ ∗ + i z ( x + H ) σ ∗ + i z − x σ ∗ + i z = ∫ x x + H y σ ∗ + i z − 1 d y . Hence, when s = 0 s=0 s = 0 L ( z , 0 ) \mathcal{L}(z,0) L ( z , 0 ) K x ( − z ) K_x(-z) K x ( − z )
L ( z , 0 ) = 1 X ∫ R g ( x X ) K x ( z ) K x ( − z ) d x = 1 X ∫ R g ( x X ) ∣ K x ( z ) ∣ 2 d x , \mathcal{L}(z,0)=\frac{1}{X}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)K_x(z)K_x(-z)\,dx=\frac{1}{X}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)|K_x(z)|^2\,dx, L ( z , 0 ) = X 1 ∫ R g ( X x ) K x ( z ) K x ( − z ) d x = X 1 ∫ R g ( X x ) ∣ K x ( z ) ∣ 2 d x , because K x ( − z ) = K x ( z ) ‾ K_x(-z)=\overline{K_x(z)} K x ( − z ) = K x ( z ) z z z [31 Theorem 71, pp. 94–95] ,
1 2 π ∫ R ∣ K x ( z ) ∣ 2 d z = ∫ x x + H y 2 σ ∗ − 1 d y = H x 2 σ ∗ − 1 + O ( H h x 2 σ ∗ − 1 ) \frac{1}{2\pi}\int_{\mathbb{R}}|K_x(z)|^2\,dz=\int_x^{x+H}y^{2\sigma_*-1}\,dy=Hx^{2\sigma_*-1}+O(Hhx^{2\sigma_*-1}) 2 π 1 ∫ R ∣ K x ( z ) ∣ 2 d z = ∫ x x + H y 2 σ ∗ − 1 d y = H x 2 σ ∗ − 1 + O ( H h x 2 σ ∗ − 1 ) and consequently, by setting x = X v x=Xv x = X v
1 2 π ∫ R L ( z , 0 ) d z = 1 X ∫ R g ( x X ) ∫ x x + H y 2 σ ∗ − 1 d y d x = H X 2 σ ∗ − 1 ∫ 0 ∞ g ( v ) v 2 σ ∗ − 1 d v + O ( H h X 2 σ ∗ − 1 ) , \frac{1}{2\pi}\int_{\mathbb{R}}\mathcal{L}(z,0)\,dz=\frac{1}{X}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)\int_x^{x+H}y^{2\sigma_*-1}\,dy\,dx
=HX^{2\sigma_*-1}\int_0^\infty g(v)v^{2\sigma_*-1}\,dv+O(HhX^{2\sigma_*-1}), 2 π 1 ∫ R L ( z , 0 ) d z = X 1 ∫ R g ( X x ) ∫ x x + H y 2 σ ∗ − 1 d y d x = H X 2 σ ∗ − 1 ∫ 0 ∞ g ( v ) v 2 σ ∗ − 1 d v + O ( H h X 2 σ ∗ − 1 ) , since x ≍ X x\asymp X x ≍ X H = o ( X ) H=o(X) H = o ( X ) X 2 σ ∗ − 1 = X 2 / log T = O ( 1 ) X^{2\sigma_*-1}=X^{2/\log T}=O(1) X 2 σ ∗ − 1 = X 2/ l o g T = O ( 1 ) v 2 σ ∗ − 1 = O g ( 1 ) v^{2\sigma_*-1}=O_g(1) v 2 σ ∗ − 1 = O g ( 1 ) g g g o ( H ) o(H) o ( H ) K x ( z ) K_x(z) K x ( z ) x ≍ X x\asymp X x ≍ X g ( x / X ) g(x/X) g ( x / X )
∣ K x ( z ) ∣ ≤ ∫ x x + H y σ ∗ − 1 d y ≪ H X σ ∗ − 1 . |K_x(z)|\leq\int_x^{x+H}y^{\sigma_*-1}\,dy\ll HX^{\sigma_*-1}. ∣ K x ( z ) ∣ ≤ ∫ x x + H y σ ∗ − 1 d y ≪ H X σ ∗ − 1 . The quotient representation of K x ( z ) K_x(z) K x ( z ) ∣ z ∣ ≥ 1 |z|\geq1 ∣ z ∣ ≥ 1
∣ K x ( z ) ∣ ≤ ( x + H ) σ ∗ + x σ ∗ ∣ σ ∗ + i z ∣ ≪ X σ ∗ ∣ z ∣ . |K_x(z)|\leq\frac{(x+H)^{\sigma_*}+x^{\sigma_*}}{|\sigma_*+iz|}\ll\frac{X^{\sigma_*}}{|z|}. ∣ K x ( z ) ∣ ≤ ∣ σ ∗ + i z ∣ ( x + H ) σ ∗ + x σ ∗ ≪ ∣ z ∣ X σ ∗ . Using the definition of L ( z , 0 ) \mathcal{L}(z,0) L ( z , 0 )
L ( z , 0 ) ≪ H 2 X 2 σ ∗ − 2 , \mathcal{L}(z,0)\ll H^2X^{2\sigma_*-2}, L ( z , 0 ) ≪ H 2 X 2 σ ∗ − 2 , for all real z z z
L ( z , 0 ) ≪ X 2 σ ∗ ∣ z ∣ − 2 , \mathcal{L}(z,0) \ll X^{2\sigma_*}|z|^{-2}, L ( z , 0 ) ≪ X 2 σ ∗ ∣ z ∣ − 2 , for ∣ z ∣ ≥ 1 |z| \ge1 ∣ z ∣ ≥ 1
∫ ∣ z ∣ < 1 / ( h Z ) L ( z , 0 ) d z ≪ H 2 X 2 σ ∗ − 2 h Z = H X 2 κ Z , \int_{|z|<1/(hZ)} \mathcal{L}(z,0)\,dz \ll\frac{H^2X^{2\sigma_*-2}}{hZ}=\frac{HX^{2\kappa}}{Z}, ∫ ∣ z ∣ < 1/ ( h Z ) L ( z , 0 ) d z ≪ h Z H 2 X 2 σ ∗ − 2 = Z H X 2 κ , ∫ ∣ z ∣ > Z / h L ( z , 0 ) d z ≪ X 2 σ ∗ ∫ Z / h ∞ z − 2 d z ≪ h X 2 σ ∗ Z = H X 2 κ Z . \int_{|z|>Z/h} \mathcal{L}(z,0)\,dz \ll X^{2\sigma_*}\int_{Z/h}^{\infty}z^{-2}\,dz \ll\frac{hX^{2\sigma_*}}{Z}=\frac{HX^{2\kappa}}{Z}. ∫ ∣ z ∣ > Z / h L ( z , 0 ) d z ≪ X 2 σ ∗ ∫ Z / h ∞ z − 2 d z ≪ Z h X 2 σ ∗ = Z H X 2 κ . Since X 2 κ = O ( 1 ) X^{2\kappa}=O(1) X 2 κ = O ( 1 ) Z → ∞ Z\to\infty Z → ∞ o ( H ) o(H) o ( H )
1 h Z ≤ ∣ z ∣ ≤ Z h \frac{1}{hZ}\le|z|\le\frac{Z}{h} h Z 1 ≤ ∣ z ∣ ≤ h Z with an error o ( H ) o(H) o ( H )
1 2 π H ∫ 1 h Z ≤ ∣ z ∣ ≤ Z h L ( z , 0 ) d z = X 2 κ ∫ 0 ∞ g ( v ) v 2 κ d v + o ( 1 ) = X 2 κ ( ∫ 0 ∞ g ( v ) d v + o ( 1 ) ) . \frac{1}{2\pi H}\int_{\frac{1}{hZ}\le|z|\le\frac{Z}{h}}\mathcal{L}(z,0)\,dz
=X^{2\kappa}\int_0^\infty g(v)v^{2\kappa}\,dv+o(1)
=X^{2\kappa}\left(\int_0^\infty g(v)\,dv+o(1)\right). 2 π H 1 ∫ h Z 1 ≤ ∣ z ∣ ≤ h Z L ( z , 0 ) d z = X 2 κ ∫ 0 ∞ g ( v ) v 2 κ d v + o ( 1 ) = X 2 κ ( ∫ 0 ∞ g ( v ) d v + o ( 1 ) ) . as we have v 2 κ = 1 + o ( 1 ) v^{2\kappa}=1+o(1) v 2 κ = 1 + o ( 1 ) supp g \operatorname{supp}g supp g g g g ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) κ = 1 / log T = o ( 1 ) \kappa=1/\log T=o(1) κ = 1/ log T = o ( 1 )
d [ Y k ! 2 ( 2 κ + i s ) − k 2 W k ( 2 κ + i s ; Y ) ] . d\left[\frac{Y}{k!^2}(2\kappa+is)^{-k^2}\mathcal{W}_k(2\kappa+is;Y)\right]. d [ k ! 2 Y ( 2 κ + i s ) − k 2 W k ( 2 κ + i s ; Y ) ] . By the product rule, (253) and (254) give
d d Y [ Y k ! 2 ( 2 κ + i s ) − k 2 W k ( 2 κ + i s ; Y ) ] = 1 k ! 2 ( 2 κ + i s ) − k 2 W k ( 2 κ + i s ; 1 / h ) + O ( ( 1 + log Z ) ( log X ) 2 k − 1 ) . (255) \begin{aligned}
\frac{d}{dY}\left[\frac{Y}{k!^2}(2\kappa+is)^{-k^2}\mathcal{W}_k(2\kappa+is;Y)\right]
&=\frac{1}{k!^2}(2\kappa+is)^{-k^2}\mathcal{W}_k(2\kappa+is;1/h)\\
&\quad+O\left((1+\log Z)(\log X)^{2k-1}\right). \tag*{(255)}
\end{aligned} d Y d [ k ! 2 Y ( 2 κ + i s ) − k 2 W k ( 2 κ + i s ; Y ) ] = k ! 2 1 ( 2 κ + i s ) − k 2 W k ( 2 κ + i s ; 1/ h ) + O ( ( 1 + log Z ) ( log X ) 2 k − 1 ) . (255) The representation of L ( z , 0 ) \mathcal{L}(z,0) L ( z , 0 )
∫ 1 h Z ≤ ∣ z ∣ ≤ Z / h L ( z , 0 ) d z ≪ H . \int_{\frac{1}{hZ}\le|z|\le Z/h}\mathcal{L}(z,0)\,dz\ll H. ∫ h Z 1 ≤ ∣ z ∣ ≤ Z / h L ( z , 0 ) d z ≪ H . After including the outside factor 1 / H 1/H 1/ H ∣ s ∣ ≤ ( log X ) − 1 + ε |s|\le(\log X)^{-1+\varepsilon} ∣ s ∣ ≤ ( log X ) − 1 + ε (255) contributes
≪ ( 1 + log Z ) ( log X ) 2 k − 2 + ε = o ( ( log X ) 2 k − 1 ) . \ll(1+\log Z)(\log X)^{2k-2+\varepsilon}=o\left((\log X)^{2k-1}\right). ≪ ( 1 + log Z ) ( log X ) 2 k − 2 + ε = o ( ( log X ) 2 k − 1 ) . Thus the normalised conjectural main term may be evaluated at Y = 1 / h = X / H Y=1/h=X/H Y = 1/ h = X / H (246) , with X H = X 1 / α \frac{X}{H}=X^{1/\alpha} H X = X 1/ α z z z
( ∫ 0 ∞ g ( v ) d v + o ( 1 ) ) 1 2 π 1 k ! 2 ( 2 π i ) 2 k ∫ ∣ s ∣ ≤ ( log X ) − 1 + ε ( X 1 / α ) α ( 2 κ + i s ) × ∫ C u 1 ⋯ ∫ C u k ( X 1 / α ) 1 2 ( ∑ j = 1 k u j − ∑ ℓ = 1 k v ℓ ) ∏ j , ℓ = 1 k ( v ℓ − u j − 2 κ − i s ) ∏ i < j ( u i − u j ) 2 ∏ i < j ( v i − v j ) 2 × ( ∑ j = 1 k 1 u j + ∑ ℓ = 1 k 1 v ℓ − 2 κ − i s − 1 2 log X 1 / α ) k ( ∑ j = 1 k 1 u j + 2 κ + i s + ∑ ℓ = 1 k 1 v ℓ + 1 2 log X 1 / α ) k × ( 2 κ + i s ) − k 2 ∏ j = 1 k u j − k d u j ∏ ℓ = 1 k v ℓ − k d v ℓ d s + o ( ( log X ) 2 k − 1 ) . (256) \begin{aligned}
&\left(\int_0^\infty g(v)\,dv+o(1)\right)\frac{1}{2\pi}\frac{1}{k!^2(2\pi i)^{2k}}
\int_{|s|\le(\log X)^{-1+\varepsilon}}\left(X^{1/\alpha}\right)^{\alpha(2\kappa+is)}
\\
&\quad\times\int_{\mathcal{C}_{u_1}}\cdots\int_{\mathcal{C}_{u_k}}
\left(X^{1/\alpha}\right)^{\frac{1}{2}\left(\sum_{j=1}^k u_j-\sum_{\ell=1}^k v_\ell\right)}
\prod_{j,\ell=1}^k(v_\ell-u_j-2\kappa-is)
\prod_{i<j}(u_i-u_j)^2
\prod_{i<j}(v_i-v_j)^2
\\
&\quad\times\left(\sum_{j=1}^k\frac{1}{u_j}+\sum_{\ell=1}^k\frac{1}{v_\ell-2\kappa-is}-\frac{1}{2}\log X^{1/\alpha}\right)^k
\left(\sum_{j=1}^k\frac{1}{u_j+2\kappa+is}+\sum_{\ell=1}^k\frac{1}{v_\ell}+\frac{1}{2}\log X^{1/\alpha}\right)^k
\\
&\quad\times(2\kappa+is)^{-k^2}\prod_{j=1}^k u_j^{-k}\,du_j\prod_{\ell=1}^k v_\ell^{-k}\,dv_\ell\,ds
+o\left((\log X)^{2k-1}\right). \tag*{(256)}
\end{aligned} ( ∫ 0 ∞ g ( v ) d v + o ( 1 ) ) 2 π 1 k ! 2 ( 2 π i ) 2 k 1 ∫ ∣ s ∣ ≤ ( l o g X ) − 1 + ε ( X 1/ α ) α ( 2 κ + i s ) × ∫ C u 1 ⋯ ∫ C u k ( X 1/ α ) 2 1 ( ∑ j = 1 k u j − ∑ ℓ = 1 k v ℓ ) j , ℓ = 1 ∏ k ( v ℓ − u j − 2 κ − i s ) i < j ∏ ( u i − u j ) 2 i < j ∏ ( v i − v j ) 2 × ( j = 1 ∑ k u j 1 + ℓ = 1 ∑ k v ℓ − 2 κ − i s 1 − 2 1 log X 1/ α ) k ( j = 1 ∑ k u j + 2 κ + i s 1 + ℓ = 1 ∑ k v ℓ 1 + 2 1 log X 1/ α ) k × ( 2 κ + i s ) − k 2 j = 1 ∏ k u j − k d u j ℓ = 1 ∏ k v ℓ − k d v ℓ d s + o ( ( log X ) 2 k − 1 ) . (256) We make the change of variables w = 2 κ + i s w = 2\kappa+ is w = 2 κ + i s d w = i d s dw = i\,ds d w = i d s s s s
R e w = 2 κ , ∣ I m w ∣ ≤ ( log X ) − 1 + ε . \mathop{\rm Re} w = 2\kappa,\qquad|\mathop{\rm Im} w| \le(\log X)^{-1+\varepsilon}. Re w = 2 κ , ∣ Im w ∣ ≤ ( log X ) − 1 + ε . We move this segment to R e w = 1 / log X \mathop{\rm Re} w = 1/\log X Re w = 1/ log X Lemma 2.7 and Lemma 2.6 termwise on the finite segment. The factorisation (162) , with T T T X 1 / α X^{1/\alpha} X 1/ α (165) –(169) , shows that the terms with a < r a<r a < r ( r , a ) (r,a) ( r , a )
≪ k { ( log X 1 / α ) k ∣ w ∣ − k , r = 0 , k ≥ 2 , ( log X 1 / α ) 2 k − 2 r ∣ w ∣ − 2 r , 1 ≤ r ≤ k , (257) \ll_k
\begin{cases}
(\log X^{1/\alpha})^k |w|^{-k}, & r=0,\quad k\ge2,\\
(\log X^{1/\alpha})^{2k-2r}|w|^{-2r}, & 1\le r\le k,
\end{cases} \tag*{(257)} ≪ k { ( log X 1/ α ) k ∣ w ∣ − k , ( log X 1/ α ) 2 k − 2 r ∣ w ∣ − 2 r , r = 0 , k ≥ 2 , 1 ≤ r ≤ k , (257) whenever ∣ I m w ∣ ≥ ( log X ) − 1 + ε |\mathop{\rm Im}w|\ge(\log X)^{-1+\varepsilon} ∣ Im w ∣ ≥ ( log X ) − 1 + ε R e w ≍ 1 / log X \mathop{\rm Re}w\asymp1/\log X Re w ≍ 1/ log X k = 1 k=1 k = 1 r = 0 r=0 r = 0 J ~ 0 ( 0 ) ( w ; X 1 / α ) = 0 \widetilde{J}^{(0)}_0(w;X^{1/\alpha})=0 J 0 ( 0 ) ( w ; X 1/ α ) = 0 J ~ 0 ( 1 ) ( w ; X 1 / α ) = 1 \widetilde{J}^{(1)}_0(w;X^{1/\alpha})=1 J 0 ( 1 ) ( w ; X 1/ α ) = 1 O ( ∣ w ∣ − 2 ) O(|w|^{-2}) O ( ∣ w ∣ − 2 )
It follows from (257) that the omitted parts of the vertical line have total contribution
≪ k { ( log X 1 / α ) k ( log X ) ( 1 − ε ) ( k − 1 ) , r = 0 , k ≥ 2 , ( log X 1 / α ) 2 k − 2 r ( log X ) ( 1 − ε ) ( 2 r − 1 ) , 1 ≤ r ≤ k , ( log X ) 1 − ε , r = 0 , k = 1 , = o ( ( log X ) 2 k − 1 ) . \ll_k
\begin{cases}
(\log X^{1/\alpha})^k(\log X)^{(1-\varepsilon)(k-1)}, & r=0,\quad k\ge2,\\
(\log X^{1/\alpha})^{2k-2r}(\log X)^{(1-\varepsilon)(2r-1)}, & 1\le r\le k,\\
(\log X)^{1-\varepsilon}, & r=0,\quad k=1,
\end{cases}
= o\bigl((\log X)^{2k-1}\bigr). ≪ k ⎩ ⎨ ⎧ ( log X 1/ α ) k ( log X ) ( 1 − ε ) ( k − 1 ) , ( log X 1/ α ) 2 k − 2 r ( log X ) ( 1 − ε ) ( 2 r − 1 ) , ( log X ) 1 − ε , r = 0 , k ≥ 2 , 1 ≤ r ≤ k , r = 0 , k = 1 , = o ( ( log X ) 2 k − 1 ) . The horizontal segments introduced when moving R e w = 2 κ \mathop{\rm Re}w=2\kappa Re w = 2 κ R e w = 1 / log X \mathop{\rm Re}w=1/\log X Re w = 1/ log X O ( 1 / log X ) O(1/\log X) O ( 1/ log X ) (257) at ∣ I m w ∣ = ( log X ) − 1 + ε |\mathop{\rm Im}w|=(\log X)^{-1+\varepsilon} ∣ Im w ∣ = ( log X ) − 1 + ε o ( ( log X ) 2 k − 1 ) o((\log X)^{2k-1}) o (( log X ) 2 k − 1 ) w = 0 w=0 w = 0 R e w = 1 / log X \mathop{\rm Re}w=1/\log X Re w = 1/ log X o ( ( log X ) 2 k − 1 ) o((\log X)^{2k-1}) o (( log X ) 2 k − 1 ) w w w s s s (256) becomes
( ∫ 0 ∞ g ( v ) d v + o ( 1 ) ) 1 k ! 2 ( 2 π i ) 2 k + 1 ∫ R e s = 1 / log X s − k 2 ∫ C u k ∫ C v k ( X 1 / α ) α s + 1 2 ( ∑ j = 1 k u j − ∑ ℓ = 1 k v ℓ ) \left(\int_0^\infty g(v)\,dv+o(1)\right)
\frac{1}{k!^2(2\pi i)^{2k+1}}
\int_{\mathop{\rm Re}s=1/\log X}s^{-k^2}
\int_{\mathcal C_u^k}\int_{\mathcal C_v^k}
(X^{1/\alpha})^{\alpha s+\frac12\left(\sum_{j=1}^k u_j-\sum_{\ell=1}^k v_\ell\right)} ( ∫ 0 ∞ g ( v ) d v + o ( 1 ) ) k ! 2 ( 2 π i ) 2 k + 1 1 ∫ Re s = 1/ l o g X s − k 2 ∫ C u k ∫ C v k ( X 1/ α ) α s + 2 1 ( ∑ j = 1 k u j − ∑ ℓ = 1 k v ℓ ) × ( ∑ j = 1 k 1 u j + ∑ ℓ = 1 k 1 v ℓ − s − 1 2 log X 1 / α ) k ( ∑ j = 1 k 1 u j + s + ∑ ℓ = 1 k 1 v ℓ + 1 2 log X 1 / α ) k {}\times\left(\sum_{j=1}^k\frac{1}{u_j}+\sum_{\ell=1}^k\frac{1}{v_\ell-s}-\frac12\log X^{1/\alpha}\right)^k
\left(\sum_{j=1}^k\frac{1}{u_j+s}+\sum_{\ell=1}^k\frac{1}{v_\ell}+\frac12\log X^{1/\alpha}\right)^k × ( j = 1 ∑ k u j 1 + ℓ = 1 ∑ k v ℓ − s 1 − 2 1 log X 1/ α ) k ( j = 1 ∑ k u j + s 1 + ℓ = 1 ∑ k v ℓ 1 + 2 1 log X 1/ α ) k × ∏ j , ℓ = 1 k ( v ℓ − u j − s ) Δ ( u ) 2 Δ ( v ) 2 ∏ j = 1 k u j − k d u j ∏ ℓ = 1 k v ℓ − k d v ℓ d s + o ( ( log X ) 2 k − 1 ) . (258) {}\times\prod_{j,\ell=1}^k(v_\ell-u_j-s)\Delta(u)^2\Delta(v)^2
\prod_{j=1}^k u_j^{-k}\,du_j
\prod_{\ell=1}^k v_\ell^{-k}\,dv_\ell\,ds
+o\bigl((\log X)^{2k-1}\bigr). \tag*{(258)} × j , ℓ = 1 ∏ k ( v ℓ − u j − s ) Δ ( u ) 2 Δ ( v ) 2 j = 1 ∏ k u j − k d u j ℓ = 1 ∏ k v ℓ − k d v ℓ d s + o ( ( log X ) 2 k − 1 ) . (258) Here C u \mathcal C_u C u 0 0 0 − s -s − s C v \mathcal C_v C v 0 0 0 s s s Lemma 2.7 to (258) , which gives
( ∫ 0 ∞ g ( v ) d v + o ( 1 ) ) ( − 1 ) k 2 k ! 2 ( 2 π i ) 2 k + 1 ∫ R e s = 1 / log X s − k 2 ∫ C u k ∫ C v k ( X 1 / α ) α s + ∑ j = 1 k u j \left(\int_0^\infty g(v)\,dv+o(1)\right)
\frac{(-1)^{k^2}}{k!^2(2\pi i)^{2k+1}}
\int_{\mathop{\rm Re}s=1/\log X}s^{-k^2}
\int_{\mathcal C_u^k}\int_{\mathcal C_v^k}
(X^{1/\alpha})^{\alpha s+\sum_{j=1}^k u_j} ( ∫ 0 ∞ g ( v ) d v + o ( 1 ) ) k ! 2 ( 2 π i ) 2 k + 1 ( − 1 ) k 2 ∫ Re s = 1/ l o g X s − k 2 ∫ C u k ∫ C v k ( X 1/ α ) α s + ∑ j = 1 k u j × ( ∑ j = 1 k 1 u j + ∑ ℓ = 1 k 1 v ℓ − s − log X 1 / α ) k ( ∑ j = 1 k 1 u j + s + ∑ ℓ = 1 k 1 v ℓ ) k {}\times\left(\sum_{j=1}^k\frac{1}{u_j}+\sum_{\ell=1}^k\frac{1}{v_\ell-s}-\log X^{1/\alpha}\right)^k
\left(\sum_{j=1}^k\frac{1}{u_j+s}+\sum_{\ell=1}^k\frac{1}{v_\ell}\right)^k × ( j = 1 ∑ k u j 1 + ℓ = 1 ∑ k v ℓ − s 1 − log X 1/ α ) k ( j = 1 ∑ k u j + s 1 + ℓ = 1 ∑ k v ℓ 1 ) k × ∏ j , ℓ = 1 k ( u j − v ℓ + s ) Δ ( u ) 2 Δ ( v ) 2 ∏ j = 1 k u j − k d u j ∏ ℓ = 1 k v ℓ − k d v ℓ d s + o ( ( log X ) 2 k − 1 ) . (259) {}\times\prod_{j,\ell=1}^k(u_j-v_\ell+s)\Delta(u)^2\Delta(v)^2
\prod_{j=1}^k u_j^{-k}\,du_j
\prod_{\ell=1}^k v_\ell^{-k}\,dv_\ell\,ds
+o\bigl((\log X)^{2k-1}\bigr). \tag*{(259)} × j , ℓ = 1 ∏ k ( u j − v ℓ + s ) Δ ( u ) 2 Δ ( v ) 2 j = 1 ∏ k u j − k d u j ℓ = 1 ∏ k v ℓ − k d v ℓ d s + o ( ( log X ) 2 k − 1 ) . (259) In the above, we used ∏ j , ℓ = 1 k ( v ℓ − u j − s ) = ( − 1 ) k 2 ∏ j , ℓ = 1 k ( u j − v ℓ + s ) \prod_{j,\ell=1}^k(v_\ell-u_j-s)=(-1)^{k^2}\prod_{j,\ell=1}^k(u_j-v_\ell+s) ∏ j , ℓ = 1 k ( v ℓ − u j − s ) = ( − 1 ) k 2 ∏ j , ℓ = 1 k ( u j − v ℓ + s ) Lemma 2.6 is applied; for each fixed s s s R e s = 1 / log X \mathop{\rm Re}s=1/\log X Re s = 1/ log X σ = − s \sigma=-s σ = − s L = log X 1 / α L=\log X^{1/\alpha} L = log X 1/ α Φ L ( u 1 , … , u k ) = \Phi_L(u_1,\ldots,u_k)= Φ L ( u 1 , … , u k ) = ( X 1 / α ) u 1 + ⋯ + u k (X^{1/\alpha})^{u_1+\cdots+u_k} ( X 1/ α ) u 1 + ⋯ + u k Lemma 2.6 are immediate from this choice, and we obtain
( ∫ 0 ∞ g ( v ) d v + o ( 1 ) ) 1 k ! 2 ( 2 π i ) 2 k + 1 ∑ r = 0 k ( k r ) 2 ∑ a = 0 k ( k a ) ( log X 1 / α ) k − a × ∫ ℜ s = 1 / log X ( X 1 / α ) α s s − a − k J log X 1 / α , r ( a ) ( − s ) d s + o ( ( log X ) 2 k − 1 ) . (260) \left(\int_0^\infty g(v)\,dv+o(1)\right)\frac{1}{k!^2(2\pi i)^{2k+1}}\sum_{r=0}^{k}\binom{k}{r}^{2}\sum_{a=0}^{k}\binom{k}{a}\left(\log X^{1/\alpha}\right)^{k-a}
\times\int_{\Re s=1/\log X}\left(X^{1/\alpha}\right)^{\alpha s}s^{-a-k}J_{\log X^{1/\alpha},r}^{(a)}(-s)\,ds+o\left((\log X)^{2k-1}\right). \tag*{(260)} ( ∫ 0 ∞ g ( v ) d v + o ( 1 ) ) k ! 2 ( 2 π i ) 2 k + 1 1 r = 0 ∑ k ( r k ) 2 a = 0 ∑ k ( a k ) ( log X 1/ α ) k − a × ∫ ℜ s = 1/ l o g X ( X 1/ α ) α s s − a − k J l o g X 1/ α , r ( a ) ( − s ) d s + o ( ( log X ) 2 k − 1 ) . (260) where
J log X 1 / α , r ( a ) ( − s ) = ∫ c j ( r ) ∫ c ^ ℓ ( r ) ( X 1 / α ) − s ( x 1 + ⋯ + x k ) Δ ( x ) 2 Δ ( y ) 2 ∏ j , ℓ = 1 k ( x j − y ℓ − 1 ) × ( ∑ j = 1 k 1 x j + ∑ ℓ = 1 k 1 y ℓ + 1 ) a ( ∑ j = 1 k 1 x j − 1 + ∑ ℓ = 1 k 1 y ℓ ) k ∏ j = 1 k x j − k d x j ∏ ℓ = 1 k y ℓ − k d y ℓ . (261) J_{\log X^{1/\alpha},r}^{(a)}(-s)=\int_{c_j(r)}\int_{\widehat{c}_\ell(r)}\left(X^{1/\alpha}\right)^{-s(x_1+\cdots+x_k)}\Delta(x)^2\Delta(y)^2\prod_{j,\ell=1}^{k}(x_j-y_\ell-1)
\times\left(\sum_{j=1}^{k}\frac{1}{x_j}+\sum_{\ell=1}^{k}\frac{1}{y_\ell+1}\right)^a\left(\sum_{j=1}^{k}\frac{1}{x_j-1}+\sum_{\ell=1}^{k}\frac{1}{y_\ell}\right)^k\prod_{j=1}^{k}x_j^{-k}\,dx_j\prod_{\ell=1}^{k}y_\ell^{-k}\,dy_\ell. \tag*{(261)} J l o g X 1/ α , r ( a ) ( − s ) = ∫ c j ( r ) ∫ c ℓ ( r ) ( X 1/ α ) − s ( x 1 + ⋯ + x k ) Δ ( x ) 2 Δ ( y ) 2 j , ℓ = 1 ∏ k ( x j − y ℓ − 1 ) × ( j = 1 ∑ k x j 1 + ℓ = 1 ∑ k y ℓ + 1 1 ) a ( j = 1 ∑ k x j − 1 1 + ℓ = 1 ∑ k y ℓ 1 ) k j = 1 ∏ k x j − k d x j ℓ = 1 ∏ k y ℓ − k d y ℓ . (261) The contours c j ( r ) c_j(r) c j ( r ) c ^ ℓ ( r ) \widehat{c}_\ell(r) c ℓ ( r ) (39) . We now evaluate the s s s (260) . Substituting (261) into this expression, the s s s ( r , a ) (r,a) ( r , a )
( X 1 / α ) s ( α − ∑ j = 1 k x j ) s − a − k . \left(X^{1/\alpha}\right)^{s\left(\alpha-\sum_{j=1}^{k}x_j\right)}s^{-a-k}. ( X 1/ α ) s ( α − ∑ j = 1 k x j ) s − a − k . When the contour shift crosses s = 0 s=0 s = 0
Res s = 0 [ ( X 1 / α ) s ( α − ∑ j = 1 k x j ) s − a − k ] = ( log X 1 / α ) a + k − 1 ( a + k − 1 ) ! ( α − ∑ j = 1 k x j ) a + k − 1 . (262) \mathop{\operatorname{Res}}_{s=0}\left[\left(X^{1/\alpha}\right)^{s\left(\alpha-\sum_{j=1}^{k}x_j\right)}s^{-a-k}\right]=\frac{\left(\log X^{1/\alpha}\right)^{a+k-1}}{(a+k-1)!}\left(\alpha-\sum_{j=1}^{k}x_j\right)^{a+k-1}. \tag*{(262)} Res s = 0 [ ( X 1/ α ) s ( α − ∑ j = 1 k x j ) s − a − k ] = ( a + k − 1 )! ( log X 1/ α ) a + k − 1 ( α − j = 1 ∑ k x j ) a + k − 1 . (262) Multiplication by the factor ( log X 1 / α ) k − a \left(\log X^{1/\alpha}\right)^{k-a} ( log X 1/ α ) k − a (260) gives the common scale ( log X 1 / α ) 2 k − 1 \left(\log X^{1/\alpha}\right)^{2k-1} ( log X 1/ α ) 2 k − 1 α \alpha α (39) . The contours in (39) are independent of α \alpha α α \alpha α (261) agree with those in the corresponding summand of (39) . Hence, after inserting (262) into (260) and summing over a a a r r r
( log X 1 / α ) 2 k − 1 P r , k ′ ( α ) . \left(\log X^{1/\alpha}\right)^{2k-1}P'_{r,k}(\alpha). ( log X 1/ α ) 2 k − 1 P r , k ′ ( α ) . The vanishing and degree bounds (165) and (169) apply in the same way. Thus the contour shift selection rule from (180) gives a contribution only from r < α r<\alpha r < α r > α r>\alpha r > α α \alpha α r = α r=\alpha r = α (186) . Applying (262) and summing over the surviving values r < α r<\alpha r < α
1 k ! 2 ∑ 0 ≤ r < α ( k r ) 2 P r , k ′ ( α ) ( log X 1 / α ) 2 k − 1 + o ( ( log X ) 2 k − 1 ) . (263) \frac{1}{k!^2}\sum_{0\le r<\alpha}\binom{k}{r}^{2}P'_{r,k}(\alpha)\left(\log X^{1/\alpha}\right)^{2k-1}+o\left((\log X)^{2k-1}\right). \tag*{(263)} k ! 2 1 0 ≤ r < α ∑ ( r k ) 2 P r , k ′ ( α ) ( log X 1/ α ) 2 k − 1 + o ( ( log X ) 2 k − 1 ) . (263) In particular, this proves
1 H X ∫ R g ( x X ) ∣ P k ( x ; H ) ∣ 2 d x ≪ ( log X ) 2 k − 1 , \frac{1}{HX}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)|P_k(x;H)|^2\,dx\ll(\log X)^{2k-1}, H X 1 ∫ R g ( X x ) ∣ P k ( x ; H ) ∣ 2 d x ≪ ( log X ) 2 k − 1 , which in turn proves (206) . Hence, by (203) , we have
V ~ g , k ( X , H ) = 1 H X ∫ R g ( x X ) ∣ P k ( x ; H ) ∣ 2 d x + o ( ( log X ) 2 k − 1 ) . (264) \widetilde{V}_{g,k}(X,H)=\frac{1}{HX}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)|P_k(x;H)|^2\,dx+o\left((\log X)^{2k-1}\right). \tag*{(264)} V g , k ( X , H ) = H X 1 ∫ R g ( X x ) ∣ P k ( x ; H ) ∣ 2 d x + o ( ( log X ) 2 k − 1 ) . (264) Therefore, we can now write the variance with a separated dependence on g ( v ) g(v) g ( v ) (263) with the factor ∫ 0 ∞ g ( v ) d v + o ( 1 ) \int_0^\infty g(v)\,dv+o(1) ∫ 0 ∞ g ( v ) d v + o ( 1 ) (260) , the central contribution is
( ∫ 0 ∞ g ( v ) d v ) 1 k ! 2 ∑ 0 ≤ r < α ( k r ) 2 P r , k ′ ( α ) ( log X 1 / α ) 2 k − 1 + o ( ( log X ) 2 k − 1 ) . (265) \left(\int_0^\infty g(v)\,dv\right)\frac{1}{k!^2}\sum_{0\le r<\alpha}\binom{k}{r}^2 P'_{r,k}(\alpha)\left(\log X^{1/\alpha}\right)^{2k-1}+o\left((\log X)^{2k-1}\right). \tag*{(265)} ( ∫ 0 ∞ g ( v ) d v ) k ! 2 1 0 ≤ r < α ∑ ( r k ) 2 P r , k ′ ( α ) ( log X 1/ α ) 2 k − 1 + o ( ( log X ) 2 k − 1 ) . (265) Due to the established negligibility of E 1 E_1 E 1 E 2 E_2 E 2 E 3 E_3 E 3
1 H X ∫ R g ( x X ) ∣ P k ( x ; H ) ∣ 2 d x = ( ∫ 0 ∞ g ( v ) d v ) 1 k ! 2 ∑ 0 ≤ r < α ( k r ) 2 P r , k ′ ( α ) ( log X 1 / α ) 2 k − 1 + o ( ( log X ) 2 k − 1 ) . (266) \frac{1}{HX}\int_{\mathbb{R}}g\left(\frac{x}{X}\right)|\mathcal{P}_k(x;H)|^2\,dx=\left(\int_0^\infty g(v)\,dv\right)\frac{1}{k!^2}\sum_{0\le r<\alpha}\binom{k}{r}^2 P'_{r,k}(\alpha)\left(\log X^{1/\alpha}\right)^{2k-1}+o\left((\log X)^{2k-1}\right). \tag*{(266)} H X 1 ∫ R g ( X x ) ∣ P k ( x ; H ) ∣ 2 d x = ( ∫ 0 ∞ g ( v ) d v ) k ! 2 1 0 ≤ r < α ∑ ( r k ) 2 P r , k ′ ( α ) ( log X 1/ α ) 2 k − 1 + o ( ( log X ) 2 k − 1 ) . (266) Combining the above with (264) gives the desired asymptotic for V ~ g , k ( X , H ) \widetilde{V}_{g,k}(X,H) V g , k ( X , H ) (41) . In particular, we have shown (192) and, to then relate the variance to δ k ( α ) \delta_k(\alpha) δ k ( α ) (40) to write
lim X → ∞ V k ( X , X 1 − 1 / α ) ( log X 1 / α ) 2 k − 1 = D k ( α ) = δ k ( α ) . (267) \lim_{X\to\infty}\frac{\mathcal{V}_k(X,X^{1-1/\alpha})}{(\log X^{1/\alpha})^{2k-1}}=\mathcal{D}_k(\alpha)=\delta_k(\alpha). \tag*{(267)} X → ∞ lim ( log X 1/ α ) 2 k − 1 V k ( X , X 1 − 1/ α ) = D k ( α ) = δ k ( α ) . (267) As Theorem 2.2 requires stronger hypotheses for the parameter Y ( T ) Y(T) Y ( T ) D k ( α ) = M k ′ ( α ) \mathcal{D}_k(\alpha)=M'_k(\alpha) D k ( α ) = M k ′ ( α )
Proof of Theorem 2.4
Let k ≥ 4 k\ge4 k ≥ 4 α > 0 \alpha>0 α > 0 log X / log q → α \log X/\log q\to\alpha log X / log q → α q → ∞ q\to\infty q → ∞ L L L q q q Y ( q ) = q α / 2 + ε Y(q)=q^{\alpha/2+\varepsilon} Y ( q ) = q α /2 + ε ε > 0 \varepsilon>0 ε > 0 η > 0 \eta>0 η > 0 α η < ε \alpha\eta<\varepsilon α η < ε T = X 1 / 2 + η T=X^{1/2+\eta} T = X 1/2 + η T = q α / 2 + α η + o ( 1 ) T=q^{\alpha/2+\alpha\eta+o(1)} T = q α /2 + α η + o ( 1 ) 2 T ≤ Y ( q ) 2T\le Y(q) 2 T ≤ Y ( q ) q q q χ 0 \chi_0 χ 0 q q q χ m o d q \chi\bmod q χ mod q
I χ ( X ) : = ∑ n ≤ X Λ ∗ k ( n ) χ ( n ) − 1 χ = χ 0 Res s = 1 [ ( − ζ q ′ ζ q ( s ) ) k X s s ] . (268) I_\chi(X):=\sum_{n\le X}\Lambda^{*k}(n)\chi(n)-\mathbf{1}_{\chi=\chi_0}\mathop{\operatorname{Res}}_{s=1}\left[\left(-\frac{\zeta'_q}{\zeta_q}(s)\right)^k\frac{X^s}{s}\right]. \tag*{(268)} I χ ( X ) := n ≤ X ∑ Λ ∗ k ( n ) χ ( n ) − 1 χ = χ 0 Res s = 1 [ ( − ζ q ζ q ′ ( s ) ) k s X s ] . (268) For ℜ s > 1 \Re s>1 ℜ s > 1
( − L ′ L ( s , χ ) ) k = ∑ n ≥ 1 Λ ∗ k ( n ) χ ( n ) n s . (269) \left(-\frac{L'}{L}(s,\chi)\right)^k=\sum_{n\ge1}\frac{\Lambda^{*k}(n)\chi(n)}{n^s}. \tag*{(269)} ( − L L ′ ( s , χ ) ) k = n ≥ 1 ∑ n s Λ ∗ k ( n ) χ ( n ) . (269) As L ( s , χ 0 ) = ζ q ( s ) L(s,\chi_0)=\zeta_q(s) L ( s , χ 0 ) = ζ q ( s )
W k ( X ; q ) = 1 φ ( q ) ∑ χ m o d q ∣ X − 1 / 2 I χ ( X ) ∣ 2 , (270) \mathcal{W}_k(X;q)=\frac{1}{\varphi(q)}\sum_{\chi\bmod q}\left|X^{-1/2}I_\chi(X)\right|^2, \tag*{(270)} W k ( X ; q ) = φ ( q ) 1 χ mod q ∑ X − 1/2 I χ ( X ) 2 , (270) with W k ( X ; q ) \mathcal{W}_k(X;q) W k ( X ; q ) (21) . We now replace X − 1 / 2 I χ ( X ) X^{-1/2}I_\chi(X) X − 1/2 I χ ( X ) ℜ s = 1 + κ \Re s=1+\kappa ℜ s = 1 + κ κ = 1 / log q \kappa=1/\log q κ = 1/ log q T T T
X − 1 / 2 I χ ( X ) = 1 2 π i ∫ 1 + κ − i T 1 + κ + i T ( − L ′ L ( s , χ ) ) k X s − 1 / 2 s d s − 1 χ = χ 0 X − 1 / 2 Res s = 1 [ ( − ζ q ′ ζ q ( s ) ) k X s s ] + E χ , tr ( X ; T ) . (271) \begin{aligned}
X^{-1/2}I_\chi(X)
={}&\frac{1}{2\pi i}\int_{1+\kappa-iT}^{1+\kappa+iT}\left(-\frac{L'}{L}(s,\chi)\right)^k\frac{X^{s-1/2}}{s}\,ds\\
&-\mathbf{1}_{\chi=\chi_0}X^{-1/2}\mathop{\operatorname{Res}}_{s=1}\left[\left(-\frac{\zeta'_q}{\zeta_q}(s)\right)^k\frac{X^s}{s}\right]+E_{\chi,\operatorname{tr}}(X;T). \tag*{(271)}
\end{aligned} X − 1/2 I χ ( X ) = 2 π i 1 ∫ 1 + κ − i T 1 + κ + i T ( − L L ′ ( s , χ ) ) k s X s − 1/2 d s − 1 χ = χ 0 X − 1/2 Res s = 1 [ ( − ζ q ζ q ′ ( s ) ) k s X s ] + E χ , tr ( X ; T ) . (271) Under GRH, uniformly in χ m o d q \chi\bmod q χ mod q
L ′ L ( σ + i t , χ ) ≪ log ( q ( ∣ t ∣ + 3 ) ) σ − 1 2 , 1 2 < σ ≤ 2 , ∣ σ + i t − 1 ∣ ≥ κ . (272) \frac{L'}{L}(\sigma+it,\chi)\ll\frac{\log(q(|t|+3))}{\sigma-\frac{1}{2}},\qquad\frac{1}{2}<\sigma\le2,\quad|\sigma+it-1|\ge\kappa. \tag*{(272)} L L ′ ( σ + i t , χ ) ≪ σ − 2 1 log ( q ( ∣ t ∣ + 3 )) , 2 1 < σ ≤ 2 , ∣ σ + i t − 1∣ ≥ κ . (272) For the principal character, the pole contribution is O ( ∣ σ + i t − 1 ∣ − 1 ) = O ( log q ) O(|\sigma+it-1|^{-1})=O(\log q) O ( ∣ σ + i t − 1 ∣ − 1 ) = O ( log q ) (272) . For sufficiently large q q q 1 2 < ℜ s ≤ 2 \frac{1}{2}<\Re s\le2 2 1 < ℜ s ≤ 2 ∣ s − 1 ∣ ≥ κ |s-1|\ge\kappa ∣ s − 1∣ ≥ κ 0 ≤ Λ ∗ ∗ ( n ) ≤ ( log n ) k 0 \leq\Lambda^{**}(n) \leq(\log n)^k 0 ≤ Λ ∗∗ ( n ) ≤ ( log n ) k ∑ d ∣ n Λ ( d ) = log n \sum_{d\mid n}\Lambda(d)=\log n ∑ d ∣ n Λ ( d ) = log n κ \kappa κ T T T χ \chi χ q q q
E χ , t r ( X ; T ) ≪ k , α , η X − 1 / 2 ( log X ) k + X 1 / 2 + κ T ( ( log q ) log ( q T ) ) k . (273) E_{\chi,\mathrm{tr}}(X;T) \ll_{k,\alpha,\eta} X^{-1/2}(\log X)^k+\frac{X^{1/2+\kappa}}{T}\bigl((\log q)\log(qT)\bigr)^k. \tag*{(273)} E χ , tr ( X ; T ) ≪ k , α , η X − 1/2 ( log X ) k + T X 1/2 + κ ( ( log q ) log ( q T ) ) k . (273) The first term accounts for the possible endpoint and near-endpoint contributions at the sharp cutoff X X X (271) to ℜ s = 1 2 + κ \Re s=\frac{1}{2}+\kappa ℜ s = 2 1 + κ L ( s , χ ) L(s,\chi) L ( s , χ ) χ = χ 0 \chi=\chi_0 χ = χ 0 s = 1 s=1 s = 1 (271) . On the horizontal segments s = σ ± i T s=\sigma\pm iT s = σ ± i T 1 2 + κ ≤ σ ≤ 1 + κ \frac{1}{2}+\kappa\leq\sigma\leq1+\kappa 2 1 + κ ≤ σ ≤ 1 + κ (272) gives
L ′ L ( σ ± i T , χ ) ≪ ( log q ) log ( q T ) . (274) \frac{L'}{L}(\sigma\pm iT,\chi)\ll(\log q)\log(qT). \tag*{(274)} L L ′ ( σ ± i T , χ ) ≪ ( log q ) log ( q T ) . (274) Since ∣ s ∣ ≍ T |s|\asymp T ∣ s ∣ ≍ T X − 1 / 2 X^{-1/2} X − 1/2
≪ ∫ 1 2 + κ 1 + κ ( ( log q ) log ( q T ) ) k X σ − 1 / 2 T d σ ≪ X 1 / 2 + κ T ( ( log q ) log ( q T ) ) k . (275) \ll\int_{\frac{1}{2}+\kappa}^{1+\kappa}\bigl((\log q)\log(qT)\bigr)^k\frac{X^{\sigma-1/2}}{T}\,d\sigma\ll\frac{X^{1/2+\kappa}}{T}\bigl((\log q)\log(qT)\bigr)^k. \tag*{(275)} ≪ ∫ 2 1 + κ 1 + κ ( ( log q ) log ( q T ) ) k T X σ − 1/2 d σ ≪ T X 1/2 + κ ( ( log q ) log ( q T ) ) k . (275) Because T = X 1 / 2 + η T=X^{1/2+\eta} T = X 1/2 + η X κ = O α ( 1 ) X^\kappa=O_\alpha(1) X κ = O α ( 1 ) (273) and (275) are O k , α , η ( X − δ ) O_{k,\alpha,\eta}(X^{-\delta}) O k , α , η ( X − δ ) δ > 0 \delta>0 δ > 0 s = 1 2 + z s=\frac{1}{2}+z s = 2 1 + z
X − 1 / 2 I χ ( X ) = I χ , T ( X ) + R χ ( X ; T ) , R χ ( X ; T ) ≪ k , α , η X − δ , (276) X^{-1/2}I_\chi(X)=I_{\chi,T}(X)+R_\chi(X;T),\qquad R_\chi(X;T)\ll_{k,\alpha,\eta}X^{-\delta}, \tag*{(276)} X − 1/2 I χ ( X ) = I χ , T ( X ) + R χ ( X ; T ) , R χ ( X ; T ) ≪ k , α , η X − δ , (276) where
I χ , T ( X ) : = 1 2 π i ∫ κ − i T κ + i T ( − L ′ L ( 1 2 + z , χ ) ) k X z 1 2 + z d z . (277) I_{\chi,T}(X):=\frac{1}{2\pi i}\int_{\kappa-iT}^{\kappa+iT}\left(-\frac{L'}{L}\left(\frac{1}{2}+z,\chi\right)\right)^k\frac{X^z}{\frac{1}{2}+z}\,dz. \tag*{(277)} I χ , T ( X ) := 2 π i 1 ∫ κ − i T κ + i T ( − L L ′ ( 2 1 + z , χ ) ) k 2 1 + z X z d z . (277) We next remove R χ ( X ; T ) R_\chi(X;T) R χ ( X ; T )
I q : = 1 φ ( q ) ∑ χ m o d q ∣ I χ , T ( X ) ∣ 2 . (278) I_q:=\frac{1}{\varphi(q)}\sum_{\chi\bmod q}|I_{\chi,T}(X)|^2. \tag*{(278)} I q := φ ( q ) 1 χ mod q ∑ ∣ I χ , T ( X ) ∣ 2 . (278) By (270) and (276) ,
W k ( X ; q ) = 1 φ ( q ) ∑ χ m o d q ∣ I χ , T ( X ) + R χ ( X ; T ) ∣ 2 . (279) W_k(X;q)=\frac{1}{\varphi(q)}\sum_{\chi\bmod q}|I_{\chi,T}(X)+R_\chi(X;T)|^2. \tag*{(279)} W k ( X ; q ) = φ ( q ) 1 χ mod q ∑ ∣ I χ , T ( X ) + R χ ( X ; T ) ∣ 2 . (279) Using ∣ A + B ∣ 2 = ∣ A ∣ 2 + 2 ℜ ( A B ‾ ) + ∣ B ∣ 2 |A+B|^2=|A|^2+2\Re(A\overline{B})+|B|^2 ∣ A + B ∣ 2 = ∣ A ∣ 2 + 2ℜ ( A B ) + ∣ B ∣ 2
W k ( X ; q ) − I q = 2 φ ( q ) ℜ ∑ χ m o d q I χ , T ( X ) R χ ( X ; T ) ‾ + 1 φ ( q ) ∑ χ m o d q ∣ R χ ( X ; T ) ∣ 2 . (280) W_k(X;q)-I_q=\frac{2}{\varphi(q)}\Re\sum_{\chi\bmod q}I_{\chi,T}(X)\overline{R_\chi(X;T)}+\frac{1}{\varphi(q)}\sum_{\chi\bmod q}|R_\chi(X;T)|^2. \tag*{(280)} W k ( X ; q ) − I q = φ ( q ) 2 ℜ χ mod q ∑ I χ , T ( X ) R χ ( X ; T ) + φ ( q ) 1 χ mod q ∑ ∣ R χ ( X ; T ) ∣ 2 . (280) By the Cauchy–Schwarz inequality,
∣ W k ( X ; q ) − I q ∣ ≤ 2 I q 1 / 2 ( 1 φ ( q ) ∑ χ m o d q ∣ R χ ( X ; T ) ∣ 2 ) 1 / 2 + 1 φ ( q ) ∑ χ m o d q ∣ R χ ( X ; T ) ∣ 2 . (281) |W_k(X;q)-I_q|\leq2I_q^{1/2}\left(\frac{1}{\varphi(q)}\sum_{\chi\bmod q}|R_\chi(X;T)|^2\right)^{1/2}+\frac{1}{\varphi(q)}\sum_{\chi\bmod q}|R_\chi(X;T)|^2. \tag*{(281)} ∣ W k ( X ; q ) − I q ∣ ≤ 2 I q 1/2 φ ( q ) 1 χ mod q ∑ ∣ R χ ( X ; T ) ∣ 2 1/2 + φ ( q ) 1 χ mod q ∑ ∣ R χ ( X ; T ) ∣ 2 . (281) It remains only to give a crude bound for I q I_q I q z = κ + i v z=\kappa+iv z = κ + i v ∣ v ∣ ≤ T |v|\leq T ∣ v ∣ ≤ T (272) gives
L ′ L ( 1 2 + z , χ ) ≪ ( log q ) log ( q T ) ≪ α , η ( log q ) 2 . (282) \frac{L'}{L}\left(\frac{1}{2}+z,\chi\right)\ll(\log q)\log(qT)\ll_{\alpha,\eta}(\log q)^2. \tag*{(282)} L L ′ ( 2 1 + z , χ ) ≪ ( log q ) log ( q T ) ≪ α , η ( log q ) 2 . (282) Also X κ = O α ( 1 ) X^\kappa=O_\alpha(1) X κ = O α ( 1 )
I χ , T ( X ) ≪ ( log q ) 2 k ∫ − T T d v ∣ 1 2 + κ + i v ∣ ≪ ( log q ) 2 k + 1 . (283) I_{\chi,T}(X)\ll(\log q)^{2k}\int_{-T}^{T}\frac{dv}{\left|\frac{1}{2}+\kappa+iv\right|}\ll(\log q)^{2k+1}. \tag*{(283)} I χ , T ( X ) ≪ ( log q ) 2 k ∫ − T T 2 1 + κ + i v d v ≪ ( log q ) 2 k + 1 . (283) Therefore I q ≪ ( log q ) 4 k + 2 \mathcal{I}_q \ll(\log q)^{4k+2} I q ≪ ( log q ) 4 k + 2 (276) in (281) gives
W k ( X ; q ) = I q + o ( ( log q ) 2 k − 1 ) . (284) \mathcal{W}_k(X;q)=\mathcal{I}_q+o\left((\log q)^{2k-1}\right). \tag*{(284)} W k ( X ; q ) = I q + o ( ( log q ) 2 k − 1 ) . (284) Expanding the square in (278) , and writing the conjugate integral on the line ℜ z 2 = − κ \Re z_2=-\kappa ℜ z 2 = − κ
W k ( X ; q ) = 1 ( 2 π i ) 2 ∫ κ − i T κ + i T ∫ − κ − i T − κ + i T X z 1 − z 2 ( 1 2 + z 1 ) ( 1 2 − z 2 ) B k ( z 1 , z 2 ; q ) d z 2 d z 1 + o ( ( log q ) 2 k − 1 ) , (285) \mathcal{W}_k(X;q)=\frac{1}{(2\pi i)^2}\int_{\kappa-iT}^{\kappa+iT}\int_{-\kappa-iT}^{-\kappa+iT}\frac{X^{z_1-z_2}}{\left(\frac{1}{2}+z_1\right)\left(\frac{1}{2}-z_2\right)}\mathcal{B}_k(z_1,z_2;q)\,\mathrm{d}z_2\,\mathrm{d}z_1+o\left((\log q)^{2k-1}\right), \tag*{(285)} W k ( X ; q ) = ( 2 π i ) 2 1 ∫ κ − i T κ + i T ∫ − κ − i T − κ + i T ( 2 1 + z 1 ) ( 2 1 − z 2 ) X z 1 − z 2 B k ( z 1 , z 2 ; q ) d z 2 d z 1 + o ( ( log q ) 2 k − 1 ) , (285) where
B k ( z 1 , z 2 ; q ) : = 1 φ ( q ) ∑ χ m o d q ( L ′ L ( 1 2 + z 1 , χ ) ) k ( L ′ L ( 1 2 − z 2 , χ ‾ ) ) k . (286) \mathcal{B}_k(z_1,z_2;q):=\frac{1}{\varphi(q)}\sum_{\chi\bmod q}\left(\frac{L'}{L}\left(\frac{1}{2}+z_1,\chi\right)\right)^k\left(\frac{L'}{L}\left(\frac{1}{2}-z_2,\overline{\chi}\right)\right)^k. \tag*{(286)} B k ( z 1 , z 2 ; q ) := φ ( q ) 1 χ mod q ∑ ( L L ′ ( 2 1 + z 1 , χ ) ) k ( L L ′ ( 2 1 − z 2 , χ ) ) k . (286) On the contours in (285) , with z = z 1 − z 2 z=z_1-z_2 z = z 1 − z 2
ℜ z = 2 κ = 2 log q , ∣ ℑ z 2 ∣ ≤ T , ∣ ℑ z ∣ ≤ 2 T . (287) \Re z=2\kappa=\frac{2}{\log q},\qquad|\Im z_2|\le T,\qquad|\Im z|\le2T. \tag*{(287)} ℜ z = 2 κ = log q 2 , ∣ℑ z 2 ∣ ≤ T , ∣ℑ z ∣ ≤ 2 T . (287) Thus Conjecture [2 ] applies throughout the integration region in (285) , since 2 T ≤ Y ( q ) 2T\le Y(q) 2 T ≤ Y ( q ) q q q 0 < ξ < 1 0<\xi<1 0 < ξ < 1 k ≥ 4 k\ge4 k ≥ 4 θ k , c = k − 1 \theta_{k,c}=k-1 θ k , c = k − 1 E c o n j E_{\mathrm{conj}} E conj (285) . Writing z 1 = κ + i ( v + y ) z_1=\kappa+i(v+y) z 1 = κ + i ( v + y ) z 2 = − κ + i v z_2=-\kappa+iv z 2 = − κ + i v a = 1 2 + κ a=\frac{1}{2}+\kappa a = 2 1 + κ
∫ R d v ∣ a + i ( v + y ) ∣ ∣ a − i v ∣ ≪ log ( 2 + ∣ y ∣ ) 1 + ∣ y ∣ . (288) \int_{\mathbb{R}}\frac{\mathrm{d}v}{|a+i(v+y)||a-iv|}\ll\frac{\log(2+|y|)}{1+|y|}. \tag*{(288)} ∫ R ∣ a + i ( v + y ) ∣∣ a − i v ∣ d v ≪ 1 + ∣ y ∣ log ( 2 + ∣ y ∣ ) . (288) Here we have enlarged the inner integration interval to R \mathbb{R} R ∣ X z 1 − z 2 ∣ = X 2 κ = O α ( 1 ) |X^{z_1-z_2}|=X^{2\kappa}=O_\alpha(1) ∣ X z 1 − z 2 ∣ = X 2 κ = O α ( 1 )
∣ E c o n j ∣ ≪ ( log q ) 2 k − 1 + ξ ∫ − 2 T 2 T log ( 2 + ∣ y ∣ ) 1 + ∣ y ∣ ( 1 + min ( ∣ 2 κ + i y ∣ , 1 ) log q ) − ( k − 1 ) d y ≪ ( log q ) 2 k − 1 + ξ ( 1 log q + ( log ( 2 T ) ) 2 ( log q ) k − 1 ) ≪ ( log q ) 2 k − 2 + ξ + ( log q ) k + 2 + ξ = o ( ( log q ) 2 k − 1 ) . (289) \begin{aligned}
|E_{\mathrm{conj}}|\ll(\log q)^{2k-1+\xi}\int_{-2T}^{2T}\frac{\log(2+|y|)}{1+|y|}\left(1+\min(|2\kappa+iy|,1)\log q\right)^{-(k-1)}\,\mathrm{d}y \\
&\ll(\log q)^{2k-1+\xi}\left(\frac{1}{\log q}+\frac{(\log(2T))^2}{(\log q)^{k-1}}\right) \\
&\ll(\log q)^{2k-2+\xi}+(\log q)^{k+2+\xi}=o\left((\log q)^{2k-1}\right). \tag*{(289)}
\end{aligned} ∣ E conj ∣ ≪ ( log q ) 2 k − 1 + ξ ∫ − 2 T 2 T 1 + ∣ y ∣ log ( 2 + ∣ y ∣ ) ( 1 + min ( ∣2 κ + i y ∣ , 1 ) log q ) − ( k − 1 ) d y ≪ ( log q ) 2 k − 1 + ξ ( log q 1 + ( log q ) k − 1 ( log ( 2 T ) ) 2 ) ≪ ( log q ) 2 k − 2 + ξ + ( log q ) k + 2 + ξ = o ( ( log q ) 2 k − 1 ) . (289) The second bound follows by splitting the y y y ∣ y ∣ = 1 |y|=1 ∣ y ∣ = 1 log T ≍ log q \log T\asymp\log q log T ≍ log q k ≥ 4 k\ge4 k ≥ 4 ξ < 1 \xi<1 ξ < 1 ( z 1 , z 2 ) (z_1,z_2) ( z 1 , z 2 ) ( z , z 2 ) (z,z_2) ( z , z 2 ) z 2 z_2 z 2 (158) , with the shifted denominator ( 1 2 + z + z 2 ) ( 1 2 − z 2 ) \left(\frac{1}{2}+z+z_2\right)\left(\frac{1}{2}-z_2\right) ( 2 1 + z + z 2 ) ( 2 1 − z 2 ) z = 2 κ + i y z=2\kappa+iy z = 2 κ + i y ∣ y ∣ ≤ 2 T |y|\le2T ∣ y ∣ ≤ 2 T
1 2 π i ∫ − κ + i max ( − T , − T − y ) − κ + i min ( T , T − y ) d z 2 ( 1 2 + z + z 2 ) ( 1 2 − z 2 ) = 1 1 + z + O ( log ( q T ) T ) . (290) \frac{1}{2\pi i}\int_{-\kappa+i\max(-T,-T-y)}^{-\kappa+i\min(T,T-y)}\frac{\mathrm{d}z_2}{\left(\frac{1}{2}+z+z_2\right)\left(\frac{1}{2}-z_2\right)}=\frac{1}{1+z}+O\left(\frac{\log(qT)}{T}\right). \tag*{(290)} 2 π i 1 ∫ − κ + i m a x ( − T , − T − y ) − κ + i m i n ( T , T − y ) ( 2 1 + z + z 2 ) ( 2 1 − z 2 ) d z 2 = 1 + z 1 + O ( T log ( q T ) ) . (290) The error term in (290) gives o ( ( log q ) 2 k − 1 ) o((\log q)^{2k-1}) o (( log q ) 2 k − 1 ) z z z
W k ( X ; q ) = 1 2 π i ∫ 2 κ − 2 i T 2 κ + 2 i T X z 1 + z K k ( z ; q ) d z + o ( ( log q ) 2 k − 1 ) . (291) \mathcal{W}_k(X;q)=\frac{1}{2\pi i}\int_{2\kappa-2iT}^{2\kappa+2iT}\frac{X^z}{1+z}\mathcal{K}_k(z;q)\,\mathrm{d}z+o\left((\log q)^{2k-1}\right). \tag*{(291)} W k ( X ; q ) = 2 π i 1 ∫ 2 κ − 2 i T 2 κ + 2 i T 1 + z X z K k ( z ; q ) d z + o ( ( log q ) 2 k − 1 ) . (291) where
K k ( z ; q ) : = 1 k ! 2 ( 2 π i ) 2 k ∫ C u k ∫ C v k q 1 2 ( ∑ j = 1 k u j − ∑ ℓ = 1 k v ℓ ) Δ ( u ) 2 Δ ( v ) 2 ( ∑ j = 1 k 1 u j + ∑ ℓ = 1 k 1 v ℓ − z − log q 2 ) k ( ∑ j = 1 k 1 u j + z + ∑ ℓ = 1 k 1 v ℓ + log q 2 ) k ∏ j , ℓ = 1 k ( v ℓ − u j − z ) z − k 2 ∏ j = 1 k u j − k d u j ∏ ℓ = 1 k v ℓ − k d v ℓ . (292) \mathcal{K}_k(z;q) := \frac{1}{k!^2(2\pi i)^{2k}}\int_{\mathcal{C}_u^k}\int_{\mathcal{C}_v^k}q^{\frac{1}{2}\left(\sum_{j=1}^{k}u_j-\sum_{\ell=1}^{k}v_\ell\right)}\Delta(u)^2\Delta(v)^2
\left(\sum_{j=1}^{k}\frac{1}{u_j}+\sum_{\ell=1}^{k}\frac{1}{v_\ell-z}-\frac{\log q}{2}\right)^k
\left(\sum_{j=1}^{k}\frac{1}{u_j+z}+\sum_{\ell=1}^{k}\frac{1}{v_\ell}+\frac{\log q}{2}\right)^k
\prod_{j,\ell=1}^{k}(v_\ell-u_j-z)z^{-k^2}\prod_{j=1}^{k}u_j^{-k}\,du_j\prod_{\ell=1}^{k}v_\ell^{-k}\,dv_\ell. \tag*{(292)} K k ( z ; q ) := k ! 2 ( 2 π i ) 2 k 1 ∫ C u k ∫ C v k q 2 1 ( ∑ j = 1 k u j − ∑ ℓ = 1 k v ℓ ) Δ ( u ) 2 Δ ( v ) 2 ( j = 1 ∑ k u j 1 + ℓ = 1 ∑ k v ℓ − z 1 − 2 log q ) k ( j = 1 ∑ k u j + z 1 + ℓ = 1 ∑ k v ℓ 1 + 2 log q ) k j , ℓ = 1 ∏ k ( v ℓ − u j − z ) z − k 2 j = 1 ∏ k u j − k d u j ℓ = 1 ∏ k v ℓ − k d v ℓ . (292) The two-centre estimates (165) and (169) give
∣ K k ( 2 κ + i y ; q ) ∣ ≪ k ( log q ) 2 k ( 1 + ∣ y ∣ log q ) − 2 . (293) \left|\mathcal{K}_k(2\kappa+iy;q)\right|\ll_k(\log q)^{2k}(1+|y|\log q)^{-2}. \tag*{(293)} ∣ K k ( 2 κ + i y ; q ) ∣ ≪ k ( log q ) 2 k ( 1 + ∣ y ∣ log q ) − 2 . (293) Its absolute integral over ∣ y ∣ ≤ 2 T |y|\le2T ∣ y ∣ ≤ 2 T O k ( ( log q ) 2 k − 1 ) O_k((\log q)^{2k-1}) O k (( log q ) 2 k − 1 ) X 2 κ = O α ( 1 ) X^{2\kappa}=O_\alpha(1) X 2 κ = O α ( 1 ) log ( q T ) ≍ log q \log(qT)\asymp\log q log ( q T ) ≍ log q (290) contributes
O k , α , η ( ( log q ) 2 k T ) = o ( ( log q ) 2 k − 1 ) , (294) O_{k,\alpha,\eta}\left(\frac{(\log q)^{2k}}{T}\right)=o\left((\log q)^{2k-1}\right), \tag*{(294)} O k , α , η ( T ( log q ) 2 k ) = o ( ( log q ) 2 k − 1 ) , (294) as required in (291) . Make the changes of variables z = w / log q z=w/\log q z = w / log q u j ↦ u j / log q u_j\mapsto u_j/\log q u j ↦ u j / log q v ℓ ↦ v ℓ / log q v_\ell\mapsto v_\ell/\log q v ℓ ↦ v ℓ / log q
L : = log q , H k ( w ) : = L − 2 k K k ( w / L ; q ) . (295) L:=\log q,\qquad H_k(w):=L^{-2k}\mathcal{K}_k(w/L;q). \tag*{(295)} L := log q , H k ( w ) := L − 2 k K k ( w / L ; q ) . (295) The scaled inner integral shows that H k H_k H k q q q
∣ H k ( 2 + i t ) ∣ ≪ k ( 1 + ∣ t ∣ ) − 2 . (296) |H_k(2+it)|\ll_k(1+|t|)^{-2}. \tag*{(296)} ∣ H k ( 2 + i t ) ∣ ≪ k ( 1 + ∣ t ∣ ) − 2 . (296) Since ∣ 1 + w / L ∣ − 1 ≤ 1 |1+w/L|^{-1}\le1 ∣1 + w / L ∣ − 1 ≤ 1 ℜ w = 2 \Re w=2 ℜ w = 2 log X / L → α \log X/L\to\alpha log X / L → α
W k ( X ; q ) L 2 k − 1 = 1 2 π i ∫ 2 − 2 i T L 2 + 2 i T L e w log X / L H k ( w ) 1 + w / L d w + o ( 1 ) ⟶ 1 2 π i ∫ ( 2 ) e α w H k ( w ) d w . (297) \frac{W_k(X;q)}{L^{2k-1}}=\frac{1}{2\pi i}\int_{2-2iTL}^{2+2iTL}\frac{e^{w\log X/L}H_k(w)}{1+w/L}\,dw+o(1)
\longrightarrow\frac{1}{2\pi i}\int_{(2)}e^{\alpha w}H_k(w)\,dw. \tag*{(297)} L 2 k − 1 W k ( X ; q ) = 2 π i 1 ∫ 2 − 2 i T L 2 + 2 i T L 1 + w / L e w l o g X / L H k ( w ) d w + o ( 1 ) ⟶ 2 π i 1 ∫ ( 2 ) e α w H k ( w ) d w . (297) For non-integer α \alpha α Lemmas 2.7 and 2.6 , followed by the residue calculation in (263) , evaluate this limiting integral. The integral is continuous in α \alpha α P m , k ′ ( m ) = 0 P'_{m,k}(m)=0 P m , k ′ ( m ) = 0 1 ≤ m ≤ k 1\le m\le k 1 ≤ m ≤ k α > 0 \alpha>0 α > 0
1 2 π i ∫ ( 2 ) e α w H k ( w ) d w = 1 k ! 2 ∑ 0 ≤ r ≤ k r < α ( k r ) 2 P r , k ′ ( α ) = δ k ( α ) . (298) \frac{1}{2\pi i}\int_{(2)}e^{\alpha w}H_k(w)\,dw=\frac{1}{k!^2}\sum_{0\le r\le k\atop r<\alpha}\binom{k}{r}^{2}P'_{r,k}(\alpha)=\delta_k(\alpha). \tag*{(298)} 2 π i 1 ∫ ( 2 ) e α w H k ( w ) d w = k ! 2 1 r < α 0 ≤ r ≤ k ∑ ( r k ) 2 P r , k ′ ( α ) = δ k ( α ) . (298) Combining (297) and (298) , we obtain
W k ( X ; q ) = ( log q ) 2 k − 1 δ k ( α ) + o ( ( log q ) 2 k − 1 ) . (299) W_k(X;q)=(\log q)^{2k-1}\delta_k(\alpha)+o\left((\log q)^{2k-1}\right). \tag*{(299)} W k ( X ; q ) = ( log q ) 2 k − 1 δ k ( α ) + o ( ( log q ) 2 k − 1 ) . (299) Dividing by ( log q ) 2 k − 1 (\log q)^{2k-1} ( log q ) 2 k − 1 q → ∞ q\to\infty q → ∞ k ≥ 4 k\ge4 k ≥ 4 k = 1 , 2 , 3 k=1,2,3 k = 1 , 2 , 3 θ k , c \theta_{k,c} θ k , c Conjecture 2 satisfies θ k , c > 2 + ξ \theta_{k,c}>2+\xi θ k , c > 2 + ξ θ k , c = 2 + c \theta_{k,c}=2+c θ k , c = 2 + c 0 < ξ < c ≤ 1 0<\xi<c\le1 0 < ξ < c ≤ 1
Acknowledgments
We would like to thank the American Institute of Mathematics for hosting several visits of the authors during the early stages of this project. We would also like to thank Sandro Bettin for answering questions related to this topic. This group was supported by NSF Grant DMS-185439. The third author was supported by the Heilbronn Institute for Mathematical Research.
References
School of Mathematics, University of Bristol, [Bristol, United Kingdom]
