Introduction and statements
For a real entire L L L L L L R L d i s t R_L^{\mathrm{dist}} R L dist R L m u l t R_L^{\mathrm{mult}} R L mult L ′ L' L ′ [ 1 / 2 , 1 ] [1/2,1] [ 1/2 , 1 ] S L S_L S L L ′ L' L ′ ( 1 / 2 , 1 ) (1/2,1) ( 1/2 , 1 ) − 1 -1 − 1
The phrase “in density” always refers to the original family measure. Thus R L / H L → c R_L/H_L \to c R L / H L → c η > 0 \eta> 0 η > 0 ∣ R L / H L − c ∣ > η \lvert R_L/H_L - c\rvert> \eta ∣ R L / H L − c ∣ > η
Our constants are
c B M = e − 1 / 2 π 2 , c ( m , v ) = e − m 2 / ( 2 v ) π 2 ( v > 0 ) . (1) c_{\mathrm{BM}} = \frac{e^{-1/2}}{\pi\sqrt{2}}, \qquad c(m,v) = \frac{e^{-m^{2}/(2v)}}{\pi\sqrt{2}} \quad(v>0). \tag*{(1)} c BM = π 2 e − 1/2 , c ( m , v ) = π 2 e − m 2 / ( 2 v ) ( v > 0 ) . (1) The parameters m m m v v v sech 2 ( t / 2 ) \operatorname{sech}^{2}(t/2) sech 2 ( t /2 ) 1 / 2 1/2 1/2 − m / v -m/\sqrt{v} − m / v c ( m , v ) c(m,v) c ( m , v ) 1 / 2 1/2 1/2 − 1 / 2 -1/2 − 1/2
Unconditional arithmetic asymptotics
Let F ( X ) \mathcal{F}(X) F ( X ) D D D 1 < ∣ D ∣ ≤ X 1<|D|\le X 1 < ∣ D ∣ ≤ X χ D \chi_D χ D ∣ D ∣ |D| ∣ D ∣
Theorem 1.1 (Quadratic Dirichlet family, unconditional). For each fixed η > 0 \eta>0 η > 0 o η ( X ) o_{\eta}(X) o η ( X ) F ( X ) \mathcal{F}(X) F ( X )
max C ∈ { R d i s t , R m u l t , S } ∣ C L ( s , χ D ) log log ∣ D ∣ − c B M ∣ ≤ η . \max_{C\in\{R^{\mathrm{dist}},R^{\mathrm{mult}},S\}}\left|\frac{C_{L}(s,\chi_D)}{\log\log|D|}-c_{\mathrm{BM}}\right|\le\eta. C ∈ { R dist , R mult , S } max log log ∣ D ∣ C L ( s , χ D ) − c BM ≤ η . No hypothesis on zeros or central values is imposed.
The full signed family in this theorem requires an arithmetic argument at the prime 2 2 2 D = 8 d D=8d D = 8 d
For the Gaussian-integer family put π = 1 + i \pi=1+i π = 1 + i d ∈ Z [ i ] d\in\mathbb{Z}[i] d ∈ Z [ i ] 2 2 2 X / 2 < N ( d ) ≤ X X/2<N(d)\le X X /2 < N ( d ) ≤ X χ π 5 d \chi_{\pi^{5}d} χ π 5 d 32 N ( d ) 32N(d) 32 N ( d ) Q d = 128 N ( d ) Q_d=128N(d) Q d = 128 N ( d ) d , − d d,-d d , − d
Theorem 1.2 (Quadratic Hecke family over Q ( i ) \mathbb{Q}(i) Q ( i ) For the preceding family, the three counts R d i s t , R m u l t , S R^{\mathrm{dist}},R^{\mathrm{mult}},S R dist , R mult , S L ( s , χ π 5 d ) L(s,\chi_{\pi^{5}d}) L ( s , χ π 5 d )
( c B M + o ( 1 ) ) log log Q d (c_{\mathrm{BM}}+o(1))\log\log Q_d ( c BM + o ( 1 )) log log Q d in density. More explicitly, at each fixed relative tolerance η > 0 \eta>0 η > 0 o η ( X ) o_{\eta}(X) o η ( X )
This is a specified quadratic Hecke family, rather than all Hecke characters over Q ( i ) \mathbb{Q}(i) Q ( i )
Let H k \mathcal{H}_k H k k k k L L L 1 / 2 1/2 1/2
Theorem 1.3 (All even weights at level one, unconditional). As even k → ∞ k\to\infty k → ∞ η > 0 \eta>0 η > 0 o η ( ∣ H k ∣ ) o_{\eta}(|\mathcal{H}_k|) o η ( ∣ H k ∣ )
max C ∈ { R d i s t , R m u l t , S } ∣ C L ( s , f ) log log ( k 2 ) − c B M ∣ ≤ η . \max_{C\in\{R^{\mathrm{dist}},R^{\mathrm{mult}},S\}}\left|\frac{C_{L(s,f)}}{\log\log(k^{2})}-c_{\mathrm{BM}}\right|\le\eta. C ∈ { R dist , R mult , S } max log log ( k 2 ) C L ( s , f ) − c BM ≤ η . The measure is unweighted. Both residue classes of even k k k 4 4 4
The three further arithmetic families under GRH
The following families are considered separately; GRH for one is not used to prove a statement about another.
(i) Fix a primitive holomorphic cuspidal eigenform f f f D D D L ( s , f ⊗ χ D ) L(s, f \otimes\chi_D) L ( s , f ⊗ χ D ) Q D ≍ f D 2 Q_D \asymp_f D^2 Q D ≍ f D 2 f f f
(ii) Let K K K L K ( s ) = ζ K ( s ) / ζ ( s ) L_K(s) = \zeta_K(s)/\zeta(s) L K ( s ) = ζ K ( s ) / ζ ( s ) L L L S 3 S_3 S 3 Q K = ∣ D K ∣ Q_K = |D_K| Q K = ∣ D K ∣
(iii) Let K K K χ , χ ‾ \chi,\overline{\chi} χ , χ q q q Z K ( s ) = L ( s , χ ) L ( s , χ ‾ ) = ζ K ( s ) / ζ ( s ) Z_K(s) = L(s,\chi)L(s,\overline{\chi}) = \zeta_K(s)/\zeta(s) Z K ( s ) = L ( s , χ ) L ( s , χ ) = ζ K ( s ) / ζ ( s ) Q K = D K = q 2 Q_K = D_K = q^2 Q K = D K = q 2
In (i), the parameter is ∣ D ∣ |D| ∣ D ∣ ∣ D K ∣ |D_K| ∣ D K ∣ X / 2 < parameter ≤ X X/2 < \text{parameter} \le X X /2 < parameter ≤ X L X = log X L_X = \log X L X = log X H = log log X H = \log\log X H = log log X H Q = log log Q H_Q = \log\log Q H Q = log log Q
Theorem 1.4 (Uniform full-count estimate under family GRH). Assume GRH for the primitive L L L c = c B M c = c_{\mathrm{BM}} c = c BM c = 1 / ( π 2 ) c = 1/(\pi\sqrt{2}) c = 1/ ( π 2 ) 0 < ε < 1 / 16 0 < \varepsilon< 1/16 0 < ε < 1/16 A ε , C ε , X ε > 0 A_\varepsilon, C_\varepsilon, X_\varepsilon> 0 A ε , C ε , X ε > 0 X ≥ X ε X \ge X_\varepsilon X ≥ X ε z z z
H − 1 / 2 + ε ≤ z ≤ 1 , H^{-1/2+\varepsilon} \le z \le1, H − 1/2 + ε ≤ z ≤ 1 , the original unweighted dyadic measure satisfies
P X { max C ∈ { R d i s t , R m u l t , S } ∣ C − c H Q ∣ > A ε z H Q } ≤ C ε log H H z 2 . (2) \mathbb{P}_X\left\{\max_{C\in\{\mathrm{Rdist},\mathrm{Rmult},S\}} |C-cH_Q|>A_\varepsilon zH_Q\right\}\le C_\varepsilon\frac{\log H}{Hz^2}. \tag*{(2)} P X { C ∈ { Rdist , Rmult , S } max ∣ C − c H Q ∣ > A ε z H Q } ≤ C ε H z 2 log H . (2) The constants are independent of z z z
The quantifier in (2) allows a choice z = z ( X ) z=z(X) z = z ( X ) z z z
For fixed η > 0 \eta>0 η > 0 O η ( log H / H ) O_\eta(\log H/H) O η ( log H / H ) η \eta η
∣ C − c H Q ∣ ≤ A H Q 2 / 3 ( log H Q ) 1 / 3 , exception O ( ( log H / H ) 1 / 3 ) , (3) |C-cH_Q|\le AH_Q^{2/3}(\log H_Q)^{1/3},\qquad\text{exception }O((\log H/H)^{1/3}), \tag*{(3)} ∣ C − c H Q ∣ ≤ A H Q 2/3 ( log H Q ) 1/3 , exception O (( log H / H ) 1/3 ) , (3) ∣ C − c H Q ∣ ≤ A ε H Q 1 / 2 + ε , exception O ε ( ( log H ) H − 2 ε ) . (4) |C-cH_Q|\le A_\varepsilon H_Q^{1/2+\varepsilon},\qquad\text{exception }O_\varepsilon((\log H)H^{-2\varepsilon}). \tag*{(4)} ∣ C − c H Q ∣ ≤ A ε H Q 1/2 + ε , exception O ε (( log H ) H − 2 ε ) . (4) These corollaries also hold for cumulative cutoffs: the exceptional counts are the displayed proportions times X X X X \sqrt{X} X z ( X ) z(X) z ( X )
The coefficientwise twist ∑ n ≥ 1 λ f ( n ) χ D ( n ) n − s \sum_{n\ge1}\lambda_f(n)\chi_D(n)n^{-s} ∑ n ≥ 1 λ f ( n ) χ D ( n ) n − s
Individual complex cubic derivatives
The augmentation in (iii) is real. An individual cubic character instead gives a complex function on the real axis. A real zero of its derivative means a simultaneous zero of its real and imaginary parts; there is no associated sign-change statistic.
Theorem 1.5 (Complex cubic characters under cubic-family GRH). Let C ( X ) C(X) C ( X ) X / 2 < q χ ≤ X X/2<q_\chi\le X X /2 < q χ ≤ X
Assume GRH for this family. Let R χ m u l t R_{\chi}^{\mathrm{mult}} R χ mult L ′ ( s , χ ) L'(s,\chi) L ′ ( s , χ ) [ 1 / 2 , 1 ] [1/2,1] [ 1/2 , 1 ] 0 < ε < 1 / 16 0<\varepsilon<1/16 0 < ε < 1/16
P C ( X ) { R χ m u l t > A ε ( log log q χ ) 1 / 2 + ε } ≪ ε log H H 1 / 2 + ε + H − 2 log H . \mathbb{P}_{\mathcal{C}(X)}\left\{R_{\chi}^{\mathrm{mult}}>A_{\varepsilon}(\log\log q_{\chi})^{1/2+\varepsilon}\right\}\ll_{\varepsilon}\frac{\log H}{H^{1/2+\varepsilon}}+H^{-2}\log H. P C ( X ) { R χ mult > A ε ( log log q χ ) 1/2 + ε } ≪ ε H 1/2 + ε log H + H − 2 log H . In particular R χ d i s t ≤ R χ m u l t = o ( log log q χ ) R_{\chi}^{\mathrm{dist}}\le R_{\chi}^{\mathrm{mult}}=o(\log\log q_{\chi}) R χ dist ≤ R χ mult = o ( log log q χ ) ≍ X \asymp X ≍ X ≍ X \asymp\sqrt{X} ≍ X
This conclusion is an upper bound, not an asymptotic of the form “R ∼ 0 R\sim0 R ∼ 0 1 − O ( H − 3 ) 1-O(H^{-3}) 1 − O ( H − 3 )
The random theorem
Fix a class of independent variables X ( p ) ∈ { 0 , 1 , − 1 } X(p)\in\{0,1,-1\} X ( p ) ∈ { 0 , 1 , − 1 } P ( X ( p ) = ± 1 ) = θ p / 2 \mathbb{P}(X(p)=\pm1)=\theta_p/2 P ( X ( p ) = ± 1 ) = θ p /2 0 < θ ∗ ≤ θ ≤ 1 0<\theta_*\le\theta\le1 0 < θ ∗ ≤ θ ≤ 1 ∣ θ p − θ ∣ ≤ A ∗ p − ε ∗ |\theta_p-\theta|\le A_*p^{-\varepsilon_*} ∣ θ p − θ ∣ ≤ A ∗ p − ε ∗ ℜ s > 1 / 2 \Re s>1/2 ℜ s > 1/2
G E ( t ) = − e − t L E ′ L E ( 1 / 2 + e − t ) . G_E(t)=-e^{-t}\frac{L'_E}{L_E}(1/2+e^{-t}). G E ( t ) = − e − t L E L E ′ ( 1/2 + e − t ) . Let N E ∙ ( u , T ) N_E^{\bullet}(u,T) N E ∙ ( u , T ) [ u , u + T ] [u,u+T] [ u , u + T ]
Theorem 1.6 (Uniform random mean and exponential concentration). Uniformly in the fixed class, deterministic u ≥ log 2 u\ge\log2 u ≥ log 2 and T ≥ 1 T\ge1 T ≥ 1
E N E ∙ ( u , T ) = e − θ / 2 π 2 T + O ( e − 5 u / 56 ) , ∙ ∈ { d i s t , m u l t } . \mathbb{E}N_E^{\bullet}(u,T)=\frac{e^{-\theta/2}}{\pi\sqrt{2}}T+O(e^{-5u/56}),\qquad\bullet\in\{\mathrm{dist},\mathrm{mult}\}. E N E ∙ ( u , T ) = π 2 e − θ /2 T + O ( e − 5 u /56 ) , ∙ ∈ { dist , mult } . For every fixed η > 0 \eta>0 η > 0 c η , C η c_{\eta},C_{\eta} c η , C η η \eta η
P { ∣ N E ∙ ( u , T ) − e − θ / 2 π 2 T ∣ > η T } ≤ C η e − c η T , ∙ ∈ { d i s t , m u l t } . \mathbb{P}\left\{\left|N_E^{\bullet}(u,T)-\frac{e^{-\theta/2}}{\pi\sqrt{2}}T\right|>\eta T\right\}\le C_{\eta}e^{-c_{\eta}T},\qquad\bullet\in\{\mathrm{dist},\mathrm{mult}\}. P { N E ∙ ( u , T ) − π 2 e − θ /2 T > η T } ≤ C η e − c η T , ∙ ∈ { dist , mult } . What is averaged and where the proof changes scale
The prime variables are independent in the model. Members of an arithmetic family are not independent random samples of an Euler product. The connection is a finite-polynomial orthogonality estimate with a power saving in the family parameter. We apply it to a finite Taylor polynomial of a characteristic function; its degree and its Euler cutoff are chosen so that their product is o ( log X ) o(\log X) o ( log X )
The variable t = − log ( σ − 1 / 2 ) t=-\log(\sigma-1/2) t = − log ( σ − 1/2 ) L ′ / L L'/L L ′ / L
For the unconditional families, the role of GRH is replaced by specifically proved density and mollified moment estimates. Their uniform ranges are kept separate. In particular, a power loss X ε X^{\varepsilon} X ε log X \log X log X
Figure 1. The proof regions for the quantitative GRH theorem, with L X = log X L_X=\log X L X = log X ν = H 1 / 2 + ε / 4 \nu=H^{1/2+\varepsilon/4} ν = H 1/2 + ε /4 σ = 1 / 2 \sigma=1/2 σ = 1/2 t = + ∞ t=+\infty t = + ∞ s s s
Figure 2. The logical inputs to the full-count results. This schematic separates the arithmetic transfer from the analytic control of central zeros. The unconditional families supply their own density and mollified moment estimates; finite-word averages alone do not replace those inputs.
Relation to earlier results and limitations
Baker and Montgomery formulated the expected log log order of the oscillation of quadratic L L L [5 ] . Their formulation includes a broader character convention; our main statement uses primitive characters indexed by fundamental discriminants. The Fekete-polynomial problem of Conrey, Granville, Poonen and Soundararajan [9 ] is related through character sums, but its unit-circle zero constant is not the Rice constant in (1) .
Related frameworks for using local splitting statistics in Artin-family moment problems appear in Cho and Kim [7 ] , and for conductor-ordered cyclic extensions in Cho and Oh [8 ] . These works provided background for the extensions considered here. The quantitative field and oriented-character inputs used in our proofs are established separately in § 6 .
None of the arithmetic statements here asserts a central nonvanishing theorem, a central simplicity theorem, a full-count central limit theorem, or a family mean asymptotic. The quantitative conductor thresholds are not made effective. The constants arising from fixed-anchor covariance matrices can be very large. Actual low-conductor computations therefore test local identities and sign brackets, not the asymptotic Rice constant. Removing GRH for the three further families requires new estimates in the central mesoscopic range; currently available estimates with conductor-power losses do not supply those inputs by direct substitution.
Random Euler products and arithmetic transfer
All random variables indexed by distinct primes are independent unless a different dependence is explicitly specified. Constants are uniform in the parameters appearing in a statement; dependence on a fixed number field is allowed. We use closed intervals when counting real zeros, and count a zero at an endpoint once.
Table 1. Comparison of the strength and setting of the statements. The intervals and counting conventions are not identified across different works. The quadratic lower bound uses [18 Theorem 1.8] , version 2; the upper bound uses [20 Theorem 1.2] , version 2.
Analytic construction and Gaussian comparison
We first give a construction which also applies to the prime ideals of a fixed number field. Write q p = N p q_{\mathfrak p}=N\mathfrak p q p = N p
Θ K ( y ) : = ∑ q p ≤ y log q p = y + O K ( y e − c K log y ) . (5) \Theta_K(y):=\sum_{q_{\mathfrak p}\le y}\log q_{\mathfrak p}=y+O_K\left(ye^{-c_K\sqrt{\log y}}\right). \tag*{(5)} Θ K ( y ) := q p ≤ y ∑ log q p = y + O K ( y e − c K l o g y ) . (5) Repeated norms are counted with their multiplicities. For K = Q K=\mathbb{Q} K = Q [17 Theorem 1.1] : the contribution of proper prime powers to ψ ( y ) \psi(y) ψ ( y ) O ( y log 2 y ) O(\sqrt{y}\log^2 y) O ( y log 2 y ) log y \log y log y
Fix 0 < θ ∗ ≤ 1 0<\theta_{*}\le1 0 < θ ∗ ≤ 1 ε ∗ > 0 \varepsilon_{*}>0 ε ∗ > 0 A ∗ < ∞ A_{*}<\infty A ∗ < ∞
P ( X p = 1 ) = P ( X p = − 1 ) = θ p / 2 , P ( X p = 0 ) = 1 − θ p , ∣ θ p − θ ∣ ≤ A ∗ q p − ε ∗ , θ ∗ ≤ θ ≤ 1. (6) \mathbb{P}(X_{\mathfrak p}=1)=\mathbb{P}(X_{\mathfrak p}=-1)=\theta_{\mathfrak p}/2,\qquad\mathbb{P}(X_{\mathfrak p}=0)=1-\theta_{\mathfrak p},\qquad|\theta_{\mathfrak p}-\theta|\le A_{*}q_{\mathfrak p}^{-\varepsilon_{*}},\qquad\theta_{*}\le\theta\le1. \tag*{(6)} P ( X p = 1 ) = P ( X p = − 1 ) = θ p /2 , P ( X p = 0 ) = 1 − θ p , ∣ θ p − θ ∣ ≤ A ∗ q p − ε ∗ , θ ∗ ≤ θ ≤ 1. (6) Put
H X ( s ) = ∑ p ∑ k ≥ 1 X p k k q p k s , L X ( s ) = e H X ( s ) , G X ( w ) = − e − w L X ′ L X ( 1 / 2 + e − w ) . (7) H_X(s)=\sum_{\mathfrak p}\sum_{k\ge1}\frac{X_{\mathfrak p}^{k}}{kq_{\mathfrak p}^{ks}},\qquad L_X(s)=e^{H_X(s)},\qquad G_X(w)=-e^{-w}\frac{L_X'}{L_X}(1/2+e^{-w}). \tag*{(7)} H X ( s ) = p ∑ k ≥ 1 ∑ k q p k s X p k , L X ( s ) = e H X ( s ) , G X ( w ) = − e − w L X L X ′ ( 1/2 + e − w ) . (7) Lemma 2.1 (Analytic versions). There is a single event of probability one on which these functions are holomorphic in ℜ s > 1 / 2 \Re s>1/2 ℜ s > 1/2 ∣ ℑ w ∣ < π / 2 |\Im w|<\pi/2 ∣ℑ w ∣ < π /2
− L X ′ L X ( s ) = ∑ p , k ≥ 1 ( log q p ) X p k q p − k s . -\frac{L_X'}{L_X}(s)=\sum_{\mathfrak p,k\ge1}(\log q_{\mathfrak p})X_{\mathfrak p}^{k}q_{\mathfrak p}^{-ks}. − L X L X ′ ( s ) = p , k ≥ 1 ∑ ( log q p ) X p k q p − k s . The function L X L_X L X L X ′ L_X' L X ′ G X G_X G X s = 1 / 2 + e − w s=1/2+e^{-w} s = 1/2 + e − w
Proof. Let Γ \Gamma Γ ℜ s > 1 / 2 \Re s > 1/2 ℜ s > 1/2 σ 0 = min Γ ℜ s > 1 / 2 \sigma_0 = \min_{\Gamma}\Re s > 1/2 σ 0 = min Γ ℜ s > 1/2 L 2 ( Γ , ∣ d s ∣ ) L^2(\Gamma, |ds|) L 2 ( Γ , ∣ d s ∣ )
E ∥ ∑ M < q p ≤ N X q p q p − s ∥ 2 2 ≤ ∣ Γ ∣ ∑ q p > M q p − 2 σ 0 ⟶ 0. \mathbb{E}\left\|\sum_{M<q_p\le N}X_{q_p}q_p^{-s}\right\|_2^2\le|\Gamma|\sum_{q_p>M}q_p^{-2\sigma_0}\longrightarrow0. E M < q p ≤ N ∑ X q p q p − s 2 2 ≤ ∣Γ∣ q p > M ∑ q p − 2 σ 0 ⟶ 0. The last series converges by (5) . For Hilbert-valued independent centered sums S j S_j S j λ \lambda λ λ P ( max j ≤ n ∥ S j ∥ > λ ) ≤ E [ ∥ S n ∥ 1 { max ∥ S j ∥ > λ } ] \lambda\mathbb{P}(\max_{j\le n}\|S_j\|>\lambda)\le\mathbb{E}[\|S_n\|\mathbf{1}_{\{\max\|S_j\|>\lambda\}}] λ P ( max j ≤ n ∥ S j ∥ > λ ) ≤ E [ ∥ S n ∥ 1 { m a x ∥ S j ∥ > λ } ] 2 λ d λ 2\lambda\,d\lambda 2 λ d λ E max j ≤ n ∥ S j ∥ 2 ≤ 4 E ∥ S n ∥ 2 \mathbb{E}\max_{j\le n}\|S_j\|^2\le4\mathbb{E}\|S_n\|^2 E max j ≤ n ∥ S j ∥ 2 ≤ 4 E ∥ S n ∥ 2 2 − 4 j 2^{-4j} 2 − 4 j 2 − j 2^{-j} 2 − j O ( 2 − 2 j ) O(2^{-2j}) O ( 2 − 2 j ) L 2 L^2 L 2
On each smaller half-plane, all terms with k ≥ 2 k\ge2 k ≥ 2 C j ( log q p ) j q p − 2 σ 1 C_j(\log q_p)^j q_p^{-2\sigma_1} C j ( log q p ) j q p − 2 σ 1 σ 1 > 1 / 2 \sigma_1>1/2 σ 1 > 1/2
There is almost surely an active prime ideal, since eventually θ p ≥ θ ∗ / 2 \theta_p\ge\theta_*/2 θ p ≥ θ ∗ /2 p p p f 1 , … , f r f_1,\ldots,f_r f 1 , … , f r p p p M = 2 l c m ( f 1 , … , f r ) M=2\mathop{\rm lcm}(f_1,\ldots,f_r) M = 2 lcm ( f 1 , … , f r ) ( p M ) − s (p^M)^{-s} ( p M ) − s ℜ s > 1 \Re s>1 ℜ s > 1
( log p ) ∑ i : X p i ≠ 0 f i > 0. (\log p)\sum_{i:X_{\mathfrak p_i}\ne0}f_i>0. ( log p ) i : X p i = 0 ∑ f i > 0. Thus cancellation between equal norms cannot make that derivative identically zero. Uniqueness here follows directly by multiplying a Dirichlet series by its least nonzero index to the power s s s s s s d s / d w = − e − w ds/dw=-e^{-w} d s / d w = − e − w L X ( s ) L_X(s) L X ( s ) e − w e^{-w} e − w L f ′ ≠ 0 L'_f\ne0 L f ′ = 0 { 0 , ± 1 } \{0,\pm1\} { 0 , ± 1 }
The following more general model will be needed for real coefficients which need not have the symmetric three-point law. Let a p , k = a p , k ( Z p ) a_{p,k}=a_{p,k}(Z_p) a p , k = a p , k ( Z p ) ∣ a p , k ∣ ≤ D |a_{p,k}|\le D ∣ a p , k ∣ ≤ D
b p = E a p , 1 = O ( p − 1 ) , v p = Var ( a p , 1 ) , c p = E a p , 2 , b_p=\mathbb{E}a_{p,1}=O(p^{-1}),\qquad v_p=\operatorname{Var}(a_{p,1}),\qquad c_p=\mathbb{E}a_{p,2}, b p = E a p , 1 = O ( p − 1 ) , v p = Var ( a p , 1 ) , c p = E a p , 2 , ∑ p ≤ x v p log p = v x + O ( x e − c log x ) , ∑ p ≤ x c p log p = m x + O ( x e − c log x ) , v > 0. (8) \sum_{p\le x}v_p\log p=vx+O\left(xe^{-c\sqrt{\log x}}\right),\qquad
\sum_{p\le x}c_p\log p=mx+O\left(xe^{-c\sqrt{\log x}}\right),\qquad v>0. \tag*{(8)} p ≤ x ∑ v p log p = v x + O ( x e − c l o g x ) , p ≤ x ∑ c p log p = m x + O ( x e − c l o g x ) , v > 0. (8) Here m m m
F ( a ) = ∑ p ( log p ) ∑ k ≥ 1 a p , k p − k ( 1 / 2 + a ) , G ( w ) = e − w F ( e − w ) . F(a)=\sum_p(\log p)\sum_{k\ge1}a_{p,k}p^{-k(1/2+a)},\qquad G(w)=e^{-w}F(e^{-w}). F ( a ) = p ∑ ( log p ) k ≥ 1 ∑ a p , k p − k ( 1/2 + a ) , G ( w ) = e − w F ( e − w ) . Lemma 2.2 (General real model). These series have holomorphic versions on ℜ a > 0 \Re a>0 ℜ a > 0 ∣ ℑ w ∣ < π / 2 |\Im w|<\pi/2 ∣ℑ w ∣ < π /2 a , b , a,b, a , b ,
E [ a F ( a ) ] = m / 2 + O ( ∣ a ∣ ) , Cov ( a F ( a ) , b F ( b ) ) = v a b ( a + b ) 2 + O ( ∣ a b ∣ ) . (9) \mathbb{E}[aF(a)]=m/2+O(|a|),\qquad
\operatorname{Cov}(aF(a),bF(b))=\frac{vab}{(a+b)^2}+O(|ab|). \tag*{(9)} E [ a F ( a )] = m /2 + O ( ∣ a ∣ ) , Cov ( a F ( a ) , b F ( b )) = ( a + b ) 2 v ab + O ( ∣ ab ∣ ) . (9) The covariance displayed is bilinear; for the conjugated covariance replace b b b b ‾ \overline{b} b R < π / 2 R<\pi/2 R < π /2 t ≥ log 2 t\ge\log2 t ≥ log 2 q ≥ 2 q\ge2 q ≥ 2
∥ sup ∣ w − t ∣ ≤ R ∣ G ( w ) ∣ ∥ q ≤ C R q . (10) \left\|\sup_{|w-t|\le R}|G(w)|\right\|_q\le C_R\sqrt{q}. \tag*{(10)} ∣ w − t ∣ ≤ R sup ∣ G ( w ) ∣ q ≤ C R q . (10) The same conclusion holds for each fixed derivative on a smaller disc. For the symmetric model, these statements hold with primes replaced by prime ideals and m = v = θ m=v=\theta m = v = θ
Proof. Apply the boundary Hilbert-space argument above to the centered prime terms. The prime means and the terms with k ≥ 2 k\ge2 k ≥ 2 Θ ( t ) = α t + E ( t ) \Theta(t)=\alpha t+E(t) Θ ( t ) = α t + E ( t ) E ( t ) ≪ t e − c log t E(t)\ll te^{-c\sqrt{\log t}} E ( t ) ≪ t e − c l o g t
∫ 1 − ∞ t − 1 − z d Θ ( t ) = α z + α + ( 1 + z ) ∫ 1 ∞ E ( t ) t − 2 − z d t . \int_{1^-}^{\infty}t^{-1-z}\,d\Theta(t)=\frac{\alpha}{z}+\alpha+(1+z)\int_1^\infty E(t)t^{-2-z}\,dt. ∫ 1 − ∞ t − 1 − z d Θ ( t ) = z α + α + ( 1 + z ) ∫ 1 ∞ E ( t ) t − 2 − z d t . The error and every fixed derivative are bounded for ℜ z ≥ 0 \Re z\ge0 ℜ z ≥ 0 z z z ∫ 0 ∞ ( 1 + u ) j e − c u d u < ∞ \int_0^\infty(1+u)^j e^{-c\sqrt{u}}\,du<\infty ∫ 0 ∞ ( 1 + u ) j e − c u d u < ∞ z − 2 z^{-2} z − 2 ( log p ) 2 (\log p)^2 ( log p ) 2 k = 2 k=2 k = 2 k = ℓ = 1 k=\ell=1 k = ℓ = 1 C ∑ p ( log p ) 2 p − 3 / 2 C\sum_p(\log p)^2p^{-3/2} C ∑ p ( log p ) 2 p − 3/2 (9) . The perturbation θ p − θ \theta_p-\theta θ p − θ q − 1 ( log q p ) j q^{-1}(\log q_p)^j q − 1 ( log q p ) j
For clarity, the elementary bounded-variable inequality used for moments is as follows. If a centered real variable U U U d d d log E e t U \log\mathbb{E}e^{tU} log E e t U d 2 / 4 d^2/4 d 2 /4 E e t U ≤ e t 2 d 2 / 8 \mathbb{E}e^{tU}\le e^{t^2d^2/8} E e t U ≤ e t 2 d 2 /8 a F ( a ) aF(a) a F ( a ) C ∣ a ∣ ( log p ) p − 1 / 2 − ℜ a C|a|(\log p)p^{-1/2-\Re a} C ∣ a ∣ ( log p ) p − 1/2 − ℜ a L q L^q L q O ( q ) O(\sqrt q) O ( q ) R < R + < π / 2 R<R_+<\pi/2 R < R + < π /2 R R R R + R_+ R + (10) . The derivative assertion follows from Cauchy’s formula once more. □
Let W W W L 2 ( 0 , ∞ ) L^2(0,\infty) L 2 ( 0 , ∞ ) ( e j ) (e_j) ( e j ) Z j Z_j Z j W ( f ) = lim n ∑ j < n ⟨ f , e j ⟩ Z j W(f)=\lim_n\sum_{j<n}\langle f,e_j\rangle Z_j W ( f ) = lim n ∑ j < n ⟨ f , e j ⟩ Z j L 2 L^2 L 2 E W ( f ) W ( h ) = ⟨ f , h ⟩ \mathbb{E}W(f)W(h)=\langle f,h\rangle E W ( f ) W ( h ) = ⟨ f , h ⟩ f δ ( x ) = 2 δ x e − δ x f_\delta(x)=2\delta\sqrt{x}e^{-\delta x} f δ ( x ) = 2 δ x e − δ x g ( t ) = W ( f e − t ) g(t)=W(f_{e^{-t}}) g ( t ) = W ( f e − t )
E g ( t ) g ( s ) = r ( t − s ) , r ( h ) = sech 2 ( h / 2 ) , G ( t ) = m / 2 + ( v / 2 ) g ( t ) . (11) \mathbb{E}g(t)g(s)=r(t-s),\qquad r(h)=\operatorname{sech}^2(h/2),\qquad G(t)=m/2+(\sqrt v/2)g(t). \tag*{(11)} E g ( t ) g ( s ) = r ( t − s ) , r ( h ) = sech 2 ( h /2 ) , G ( t ) = m /2 + ( v /2 ) g ( t ) . (11) The next lemma proves directly the special case of the Gaussian Rice formula used here; see also [3 Theorem 3.2] for the general formula.
Lemma 2.3 (Gaussian paths and exact level intensity). The process g g g C 2 C^2 C 2 u u u I I I T T T u u u
E N u ( I ) = e − u 2 / 2 π 2 T . \mathbb{E}N_u(I)=\frac{e^{-u^2/2}}{\pi\sqrt{2}}T. E N u ( I ) = π 2 e − u 2 /2 T . Proof. Write ϕ t = f e − t \phi_t=f_{e^{-t}} ϕ t = f e − t J J J 0 ≤ k ≤ 3 0\le k\le3 0 ≤ k ≤ 3 ∂ t k ϕ t ( x ) \partial_t^k\phi_t(x) ∂ t k ϕ t ( x ) 2 e − t x e − e − t x 2e^{-t}\sqrt{x}e^{-e^{-t}x} 2 e − t x e − e − t x e − t x e^{-t}x e − t x ∫ J ∥ ∂ t k ϕ t ∥ 2 2 d t < ∞ \int_J\|\partial_t^k\phi_t\|_2^2\,dt<\infty ∫ J ∥ ∂ t k ϕ t ∥ 2 2 d t < ∞ L 2 ( Ω ; H 3 ( J ) ) L^2(\Omega;H^3(J)) L 2 ( Ω ; H 3 ( J )) ∥ h ∥ H 3 ( J ) 2 = ∑ k = 0 3 ∫ J ∣ h ( k ) ∣ 2 \|h\|_{H^3(J)}^2=\sum_{k=0}^3\int_J|h^{(k)}|^2 ∥ h ∥ H 3 ( J ) 2 = ∑ k = 0 3 ∫ J ∣ h ( k ) ∣ 2
∥ h ∥ ∞ ≤ ∣ J ∣ − 1 / 2 ∥ h ∥ 2 + ∣ J ∣ 1 / 2 ∥ h ′ ∥ 2 ( h ∈ H 1 ( J ) ) \|h\|_\infty\le|J|^{-1/2}\|h\|_2+|J|^{1/2}\|h'\|_2\qquad(h\in H^1(J)) ∥ h ∥ ∞ ≤ ∣ J ∣ − 1/2 ∥ h ∥ 2 + ∣ J ∣ 1/2 ∥ h ′ ∥ 2 ( h ∈ H 1 ( J )) follows by integrating from a point with ∣ h ( t 0 ) ∣ ≤ ∣ J ∣ − 1 / 2 ∥ h ∥ 2 |h(t_0)|\le|J|^{-1/2}\|h\|_2 ∣ h ( t 0 ) ∣ ≤ ∣ J ∣ − 1/2 ∥ h ∥ 2 C 2 C^2 C 2 sup J ∣ g ′ ′ ∣ ≤ K \sup_J |g''| \le K sup J ∣ g ′′ ∣ ≤ K u u u ∣ g ( t j ) − u ∣ ≤ K h 2 / 2 |g(t_j)-u| \le Kh^2/2 ∣ g ( t j ) − u ∣ ≤ K h 2 /2 h h h ( 2 π ) − 1 / 2 (2\pi)^{-1/2} ( 2 π ) − 1/2 O J ( K h ) O_J(Kh) O J ( K h ) h ↓ 0 h \downarrow0 h ↓ 0 K → ∞ K \to\infty K → ∞
Let C n C_n C n 2 n 2^n 2 n I I I C n ↑ N u ( I ) C_n \uparrow N_u(I) C n ↑ N u ( I ) E N u ( I ) = lim n 2 n p u ( T 2 − n ) \mathbb{E}N_u(I) = \lim_{n} 2^n p_u(T2^{-n}) E N u ( I ) = lim n 2 n p u ( T 2 − n ) U h = ( g ( 0 ) + g ( h ) ) / 2 U_h = (g(0)+g(h))/2 U h = ( g ( 0 ) + g ( h )) /2 V h = ( g ( h ) − g ( 0 ) ) / h V_h = (g(h)-g(0))/h V h = ( g ( h ) − g ( 0 )) / h ( 1 + r ( h ) ) / 2 → 1 (1+r(h))/2 \to1 ( 1 + r ( h )) /2 → 1 2 ( 1 − r ( h ) ) / h 2 → 1 / 2 2(1-r(h))/h^2 \to1/2 2 ( 1 − r ( h )) / h 2 → 1/2 ∣ U h − u ∣ < h ∣ V h ∣ / 2 |U_h-u| < h|V_h|/2 ∣ U h − u ∣ < h ∣ V h ∣/2 U h U_h U h C ∣ Z ∣ C|Z| C ∣ Z ∣ Z Z Z
lim h ↓ 0 p u ( h ) h = E ∣ N ( 0 , 1 / 2 ) ∣ e − u 2 / 2 2 π = e − u 2 / 2 π 2 . \lim_{h\downarrow0} \frac{p_u(h)}{h} = \mathbb{E}|N(0,1/2)| \frac{e^{-u^2/2}}{\sqrt{2\pi}} = \frac{e^{-u^2/2}}{\pi\sqrt{2}}. h ↓ 0 lim h p u ( h ) = E ∣ N ( 0 , 1/2 ) ∣ 2 π e − u 2 /2 = π 2 e − u 2 /2 . □
Lemma 2.4 (Smooth comparison). If t 1 , … , t d ≥ log 2 t_1,\ldots,t_d \ge\log2 t 1 , … , t d ≥ log 2 b = max j e − t j b=\max_j e^{-t_j} b = max j e − t j Φ \Phi Φ C 3 C^3 C 3 M k M_k M k k k k
∣ E Φ ( ( G ( t j ) ) ) − E Φ ( ( G ( t j ) ) ) ∣ ≤ C ( d b M 1 + d 2 b 2 M 2 + d 3 b 3 M 3 ) . (12) \left|\mathbb{E}\Phi((G(t_j))) - \mathbb{E}\Phi((\mathcal{G}(t_j)))\right| \le C(dbM_1+d^2b^2M_2+d^3b^3M_3). \tag*{(12)} ∣ E Φ (( G ( t j ))) − E Φ (( G ( t j ))) ∣ ≤ C ( d b M 1 + d 2 b 2 M 2 + d 3 b 3 M 3 ) . (12) No separation or nonsingularity condition is imposed on the times.
Proof. Center the vectors of whole prime blocks, calling them U p U_p U p C b ( log p ) p − 1 / 2 Cb(\log p)p^{-1/2} C b ( log p ) p − 1/2 ∑ p E ∥ U p ∥ 3 ≤ C d 3 b 3 \sum_p \mathbb{E}\lVert U_p\rVert^3 \le Cd^3b^3 ∑ p E ∥ U p ∥ 3 ≤ C d 3 b 3 M 3 ∥ U p ∥ 3 / 6 M_3\lVert U_p\rVert^3/6 M 3 ∥ U p ∥ 3 /6 L 2 L^2 L 2 O ( b ) O(b) O ( b ) Σ u = ( 1 − u ) Σ 0 + u Σ 1 \Sigma_u=(1-u)\Sigma_0+u\Sigma_1 Σ u = ( 1 − u ) Σ 0 + u Σ 1
d d u E Φ ( Z u ) = 1 2 ∑ j , k ( Σ 1 − Σ 0 ) j k E ∂ j k Φ ( Z u ) . \frac{d}{du}\mathbb{E}\Phi(Z_u)=\frac{1}{2}\sum_{j,k}(\Sigma_1-\Sigma_0)_{jk}\mathbb{E}\partial_{jk}\Phi(Z_u). d u d E Φ ( Z u ) = 2 1 j , k ∑ ( Σ 1 − Σ 0 ) j k E ∂ j k Φ ( Z u ) . One first adds ϵ I \epsilon I ϵ I (9) gives the second error. The prime-ideal proof is identical, with its sums evaluated by (5) . □
Uniform density from discrete prime variables
Lemma 2.5 (A two-atom density criterion). Suppose G δ = ∑ p V p , δ ( Z p ) G_\delta=\sum_p V_{p,\delta}(Z_p) G δ = ∑ p V p , δ ( Z p ) 0 < δ ≤ 1 / 2 0<\delta\le1/2 0 < δ ≤ 1/2 p p p x p , y p x_p,y_p x p , y p η > 0 \eta>0 η > 0 d p d_p d p
c δ ∣ d p ∣ ( log p ) p − 1 / 2 − δ ≤ ∣ V p , δ ( x p ) − V p , δ ( y p ) ∣ ≤ C δ ( log p ) p − 1 / 2 − δ . c\delta|d_p|(\log p)p^{-1/2-\delta}\le|V_{p,\delta}(x_p)-V_{p,\delta}(y_p)|\le C\delta(\log p)p^{-1/2-\delta}. cδ ∣ d p ∣ ( log p ) p − 1/2 − δ ≤ ∣ V p , δ ( x p ) − V p , δ ( y p ) ∣ ≤ C δ ( log p ) p − 1/2 − δ . Assume also ∑ u ≤ p ≤ 2 u d p 2 log p ≥ c u \sum_{u\le p\le2u}d_p^2\log p\ge cu ∑ u ≤ p ≤ 2 u d p 2 log p ≥ c u u u u G δ G_\delta G δ δ \delta δ
sup δ , z P ( ∣ G δ − z ∣ ≤ a ) ≤ C a , sup δ , z E ( log + ∣ G δ − z ∣ − 1 ) q ≤ C Γ ( q + 1 ) ( q > 0 ) . \sup_{\delta,z}\mathbb{P}(|G_\delta-z|\le a)\le Ca,\qquad
\sup_{\delta,z}\mathbb{E}(\log^+|G_\delta-z|^{-1})^q\le C\Gamma(q+1)\qquad(q>0). δ , z sup P ( ∣ G δ − z ∣ ≤ a ) ≤ C a , δ , z sup E ( log + ∣ G δ − z ∣ − 1 ) q ≤ C Γ ( q + 1 ) ( q > 0 ) . Proof. If ϕ p ( τ ) = E e i τ V p , δ \phi_p(\tau)=\mathbb{E}e^{i\tau V_{p,\delta}} ϕ p ( τ ) = E e i τ V p , δ D p , δ D_{p,\delta} D p , δ
1 − ∣ ϕ p ( τ ) ∣ 2 = ∑ x , y P ( Z p = x ) P ( Z p = y ) [ 1 − cos ( τ ( V p , δ ( x ) − V p , δ ( y ) ) ) ] . 1-|\phi_p(\tau)|^2=\sum_{x,y}\mathbb{P}(Z_p=x)\mathbb{P}(Z_p=y)[1-\cos(\tau(V_{p,\delta}(x)-V_{p,\delta}(y)))]. 1 − ∣ ϕ p ( τ ) ∣ 2 = x , y ∑ P ( Z p = x ) P ( Z p = y ) [ 1 − cos ( τ ( V p , δ ( x ) − V p , δ ( y )))] . Retaining the two ordered atom pairs proves, when ∣ τ D p , δ ∣ ≤ 1 |\tau D_{p,\delta}|\le1 ∣ τ D p , δ ∣ ≤ 1
∣ ϕ p ( τ ) ∣ ≤ exp { − c τ 2 δ 2 d p 2 ( log p ) 2 p − 1 − 2 δ } . |\phi_p(\tau)|\le\exp\{-c\tau^2\delta^2d_p^2(\log p)^2p^{-1-2\delta}\}. ∣ ϕ p ( τ ) ∣ ≤ exp { − c τ 2 δ 2 d p 2 ( log p ) 2 p − 1 − 2 δ } . Put s = 1 / 2 + δ s=1/2+\delta s = 1/2 + δ T = 2 + ∣ τ ∣ T=2+|\tau| T = 2 + ∣ τ ∣ B = C 1 T ( log T ) 2 B=C_1T(\log T)^2 B = C 1 T ( log T ) 2 P = max ( P 0 , B 1 / s ) P=\max(P_0,B^{1/s}) P = max ( P 0 , B 1/ s ) T T T log P ≪ log T \log P\ll\log T log P ≪ log T ( log u ) u − s (\log u)u^{-s} ( log u ) u − s u ≥ P u\ge P u ≥ P C 1 C_1 C 1 p > P p>P p > P
∑ p > P d p 2 ( log p ) 2 p − 1 − 2 δ ≥ c P − 2 δ ∑ j ≥ 0 ( log P + j log 2 ) 2 − 2 δ j . \sum_{p>P}d_p^2(\log p)^2p^{-1-2\delta}\ge cP^{-2\delta}\sum_{j\ge0}(\log P+j\log2)2^{-2\delta j}. p > P ∑ d p 2 ( log p ) 2 p − 1 − 2 δ ≥ c P − 2 δ j ≥ 0 ∑ ( log P + j log 2 ) 2 − 2 δ j . For q δ = 2 − 2 δ ∈ [ 1 / 2 , 1 ) q_\delta=2^{-2\delta}\in[1/2,1) q δ = 2 − 2 δ ∈ [ 1/2 , 1 ) δ 2 ∑ j q δ j = δ 2 q δ / ( 1 − q δ ) 2 ≥ c \delta^2\sum j q_\delta^j=\delta^2q_\delta/(1-q_\delta)^2\ge c δ 2 ∑ j q δ j = δ 2 q δ / ( 1 − q δ ) 2 ≥ c
∣ ϕ ( τ ) ∣ ≤ e − c τ 2 P − 2 δ ≤ exp { − c ∣ τ ∣ / log 2 ( 2 + ∣ τ ∣ ) } . |\phi(\tau)|\le e^{-c\tau^2P^{-2\delta}}\le\exp\{-c|\tau|/\log^2(2+|\tau|)\}. ∣ ϕ ( τ ) ∣ ≤ e − c τ 2 P − 2 δ ≤ exp { − c ∣ τ ∣/ log 2 ( 2 + ∣ τ ∣ )} . Here 2 δ / s ≤ 1 2\delta/s\le1 2 δ / s ≤ 1 ∣ τ ∣ |\tau| ∣ τ ∣ δ \delta δ q u q − 1 P ( ∣ G δ − z ∣ < e − u ) qu^{q-1}\mathbb{P}(|G_\delta-z|<e^{-u}) q u q − 1 P ( ∣ G δ − z ∣ < e − u )
Joint large and small values in the symmetric model
We return to (6) . All statements in this subsection hold for fixed K K K (5) . Define A ( t ) = max ∣ w − t ∣ ≤ 4 / 5 ∣ G X ( w ) ∣ A(t)=\max_{|w-t|\le4/5}|G_X(w)| A ( t ) = max ∣ w − t ∣ ≤ 4/5 ∣ G X ( w ) ∣
Lemma 2.6 (Joint large-value moments). For 1 / 2 1/2 1/2 t 1 , … , t d ≥ log 2 t_1,\ldots,t_d\ge\log2 t 1 , … , t d ≥ log 2 λ > 0 \lambda>0 λ > 0
E exp { λ ∑ j A ( t j ) } ≤ C λ . \mathbb{E}\exp\left\{\lambda\sum_j A(t_j)\right\}\le C^\lambda. E exp { λ j ∑ A ( t j ) } ≤ C λ . Proof. Write Z ( w ) = e − w ∑ p X p ( log q p ) q p − 1 / 2 − e − w Z(w)=e^{-w}\sum_pX_p(\log q_p)q_p^{-1/2-e^{-w}} Z ( w ) = e − w ∑ p X p ( log q p ) q p − 1/2 − e − w ∣ w − t j ∣ ≤ 9 / 10 |w-t_j|\le9/10 ∣ w − t j ∣ ≤ 9/10
C δ j ∑ p log q p q p 1 + 2 c δ j ( 1 − q p − 1 / 2 − c δ j ) ≤ C , δ j = e − t j . C\delta_j\sum_p\frac{\log q_p}{q_p^{1+2c\delta_j}(1-q_p^{-1/2-c\delta_j})}\le C,\qquad\delta_j=e^{-t_j}. C δ j p ∑ q p 1 + 2 c δ j ( 1 − q p − 1/2 − c δ j ) log q p ≤ C , δ j = e − t j . If b p ( w ) b_p(w) b p ( w ) X p X_p X p Z ( w ) Z(w) Z ( w )
∑ p ∣ b p ( w j ) b p ( w k ) ∣ ≤ C δ j δ k ( δ j + δ k ) 2 ≤ C e − ∣ t j − t k ∣ . \sum_p|b_p(w_j)b_p(w_k)|\le C\frac{\delta_j\delta_k}{(\delta_j+\delta_k)^2}\le Ce^{-|t_j-t_k|}. p ∑ ∣ b p ( w j ) b p ( w k ) ∣ ≤ C ( δ j + δ k ) 2 δ j δ k ≤ C e − ∣ t j − t k ∣ . The row sums are bounded. Since E e a X p ≤ e a 2 / 2 \mathbb{E}e^{aX_p}\le e^{a^2/2} E e a X p ≤ e a 2 /2 Z ( w j ) Z(w_j) Z ( w j ) e C λ 2 d e^{C\lambda^2d} e C λ 2 d 4 d 4^d 4 d E e λ ∑ j ∣ Z ( w j ) ∣ ≤ 4 d e C λ 2 d \mathbb{E}e^{\lambda\sum_j|Z(w_j)|}\le4^de^{C\lambda^2d} E e λ ∑ j ∣ Z ( w j ) ∣ ≤ 4 d e C λ 2 d
Lemma 2.7 (Joint small values over a fixed number field). Let K K K (5) and the prime-ideal law (6) . Distinct prime ideals, including those of equal norm, have independent variables. There are h 0 ≥ 1 h_0 \ge1 h 0 ≥ 1 a 0 > 0 a_0 > 0 a 0 > 0 C 0 C_0 C 0 h 0 h_0 h 0 t 1 , … , t d ≥ log 2 t_1,\ldots,t_d \ge\log2 t 1 , … , t d ≥ log 2
E exp { a 0 ∑ j log + ∣ G X ( t j ) ∣ − 1 } ≤ C 0 d . (13) \mathbb{E}\exp\left\{a_0\sum_j \log^{+}|G_X(t_j)|^{-1}\right\}\le C_0^d. \tag*{(13)} E exp { a 0 j ∑ log + ∣ G X ( t j ) ∣ − 1 } ≤ C 0 d . (13) The following two quantitative estimates hold. If ε 0 ≤ δ j = e − t j ≤ 1 / 2 \varepsilon_0\le\delta_j=e^{-t_j}\le1/2 ε 0 ≤ δ j = e − t j ≤ 1/2 1 1 1
exp { C d 5 log 2 ( 2 d / ε 0 ) } . (14) \exp\{Cd^5\log^2(2d/\varepsilon_0)\}. \tag*{(14)} exp { C d 5 log 2 ( 2 d / ε 0 )} . (14) If the times are h 0 h_0 h 0 Q ≥ Q 0 Q\ge Q_0 Q ≥ Q 0 δ j ≤ 1 / Q \delta_j\le1/Q δ j ≤ 1/ Q n j ≥ 1 n_j\ge1 n j ≥ 1 ∑ n j ≤ N \sum n_j\le N ∑ n j ≤ N X p X_{\mathfrak p} X p q p ≤ e Q q_{\mathfrak p}\le e^Q q p ≤ e Q
P ( ∣ G X ( t j ) ∣ ≤ e − n j ( 1 ≤ j ≤ d ) ∣ ( X p ) q p ≤ e Q ) ≤ C 1 d e − ∑ n j + C 2 d 3 ( Q + 1 ) 2 e − Q / 2 + 3 N . (15) \mathbb{P}\left(|G_X(t_j)|\le e^{-n_j}\ (1\le j\le d)\mid(X_{\mathfrak p})_{q_{\mathfrak p}\le e^Q}\right)\le C_1^d e^{-\sum n_j}+C_2d^3(Q+1)^2e^{-Q/2+3N}. \tag*{(15)} P ( ∣ G X ( t j ) ∣ ≤ e − n j ( 1 ≤ j ≤ d ) ∣ ( X p ) q p ≤ e Q ) ≤ C 1 d e − ∑ n j + C 2 d 3 ( Q + 1 ) 2 e − Q /2 + 3 N . (15) Proof. Write X p = B p R p X_{\mathfrak p}=B_{\mathfrak p}R_{\mathfrak p} X p = B p R p B B B θ p \theta_{\mathfrak p} θ p R R R
Y p ( δ ) = B p ( c q ( δ ) + R p a q ( δ ) ) , c q ( δ ) = δ log q q 1 + 2 δ − 1 , a q ( δ ) = δ ( log q ) q 1 / 2 + δ q 1 + 2 δ − 1 . Y_{\mathfrak p}(\delta)=B_{\mathfrak p}(c_q(\delta)+R_{\mathfrak p}a_q(\delta)),\qquad c_q(\delta)=\frac{\delta\log q}{q^{1+2\delta}-1},\qquad a_q(\delta)=\frac{\delta(\log q)q^{1/2+\delta}}{q^{1+2\delta}-1}. Y p ( δ ) = B p ( c q ( δ ) + R p a q ( δ )) , c q ( δ ) = q 1 + 2 δ − 1 δ log q , a q ( δ ) = q 1 + 2 δ − 1 δ ( log q ) q 1/2 + δ . First consider the Gram matrix K j k = ∫ 0 1 e − ( δ j + δ k ) v d v K_{jk}=\int_0^1e^{-(\delta_j+\delta_k)v}\,dv K j k = ∫ 0 1 e − ( δ j + δ k ) v d v λ 0 = exp { − C d 2 log ( 2 d / ε 0 ) } \lambda_0=\exp\{-Cd^2\log(2d/\varepsilon_0)\} λ 0 = exp { − C d 2 log ( 2 d / ε 0 )} x j = 1 / 2 − δ j x_j=1/2-\delta_j x j = 1/2 − δ j ( 1 − e − 1 ) ε 0 (1-e^{-1})\varepsilon_0 ( 1 − e − 1 ) ε 0
det ( e v i x j ) ≥ ∏ i < j ( v j − v i ) ∏ i < j ( x j − x i ) ∏ k = 0 d − 1 k ! ( 0 ≤ v 1 < ⋯ < v d ) . \det(e^{v_i x_j})\ge\frac{\prod_{i<j}(v_j-v_i)\prod_{i<j}(x_j-x_i)}{\prod_{k=0}^{d-1}k!}\qquad(0\le v_1<\cdots<v_d). det ( e v i x j ) ≥ ∏ k = 0 d − 1 k ! ∏ i < j ( v j − v i ) ∏ i < j ( x j − x i ) ( 0 ≤ v 1 < ⋯ < v d ) . Every generalized Vandermonde in this expansion is nonnegative: a linear combination of d d d d − 1 d-1 d − 1 v i ∈ [ ( i − 1 ) / d , ( i − 1 ) / d + 1 / ( 3 d ) ] v_i\in[(i-1)/d,(i-1)/d+1/(3d)] v i ∈ [( i − 1 ) / d , ( i − 1 ) / d + 1/ ( 3 d )]
det K ≥ e − d ( 3 d ) − d ( 2 / ( 3 d ) ) d ( d − 1 ) ( ( 1 − e − 1 ) ε 0 ) d ( d − 1 ) d ! ( ∏ k = 0 d − 1 k ! ) 2 . \det K\ge\frac{e^{-d}(3d)^{-d}(2/(3d))^{d(d-1)}((1-e^{-1})\varepsilon_0)^{d(d-1)}}{d!(\prod_{k=0}^{d-1}k!)^2}. det K ≥ d ! ( ∏ k = 0 d − 1 k ! ) 2 e − d ( 3 d ) − d ( 2/ ( 3 d ) ) d ( d − 1 ) (( 1 − e − 1 ) ε 0 ) d ( d − 1 ) . The factor e − d e^{-d} e − d e − v i / 2 e^{-v_i/2} e − v i /2 tr K ≤ d \operatorname{tr}K\le d tr K ≤ d
For the joint characteristic function, the phase difference between the atoms X p = 1 , − 1 X_{\mathfrak p}=1,-1 X p = 1 , − 1
D p = 2 ( log q p ) ∑ j ξ j δ j q p − 1 / 2 − δ j 1 − q p − 1 − 2 δ j . D_{\mathfrak p}=2(\log q_{\mathfrak p})\sum_j\frac{\xi_j\delta_jq_{\mathfrak p}^{-1/2-\delta_j}}{1-q_{\mathfrak p}^{-1-2\delta_j}}. D p = 2 ( log q p ) j ∑ 1 − q p − 1 − 2 δ j ξ j δ j q p − 1/2 − δ j . Their probabilities are eventually at least θ ∗ / 4 \theta_*/4 θ ∗ /4 e − c D p 2 e^{-cD_{\mathfrak p}^2} e − c D p 2 ∣ D p ∣ ≤ 1 |D_{\mathfrak p}|\le1 ∣ D p ∣ ≤ 1 L 0 = log ( 2 d / ε 0 ) L_0=\log(2d/\varepsilon_0) L 0 = log ( 2 d / ε 0 ) q 0 = C 3 d 4 L 0 2 q_0=C_3d^4L_0^2 q 0 = C 3 d 4 L 0 2 ∥ ξ ∥ ∞ > e q 0 / ε 0 \|\xi\|_\infty>e^{q_0}/\varepsilon_0 ∥ ξ ∥ ∞ > e q 0 / ε 0 q > q 0 q>q_0 q > q 0
d ( q + 1 ) max j ( ∣ ξ j ∣ δ j e − ( 1 / 2 + δ j ) q ) = c 0 . d(q+1)\max_j(|\xi_j|\delta_je^{-(1/2+\delta_j)q})=c_0. d ( q + 1 ) j max ( ∣ ξ j ∣ δ j e − ( 1/2 + δ j ) q ) = c 0 . The left side is strictly decreasing for q ≥ 3 q\ge3 q ≥ 3 c 0 c_0 c 0 b j = ξ j δ j e − δ j q b_j=\xi_j\delta_je^{-\delta_jq} b j = ξ j δ j e − δ j q B = max ∣ b j ∣ = c 0 e q / 2 / ( d ( q + 1 ) ) B=\max|b_j|=c_0e^{q/2}/(d(q+1)) B = max ∣ b j ∣ = c 0 e q /2 / ( d ( q + 1 )) g ( v ) = ∑ b j e − δ j v g(v)=\sum b_je^{-\delta_jv} g ( v ) = ∑ b j e − δ j v q < log q p ≤ q + 1 q<\log q_{\mathfrak p}\le q+1 q < log q p ≤ q + 1 (5) , using ∣ g ∣ + ∣ g ′ ∣ ≤ C d B |g|+|g'|\le CdB ∣ g ∣ + ∣ g ′ ∣ ≤ C d B
∑ e q < q p ≤ e q + 1 ( log q p ) 2 q p g ( log q p − q ) 2 = ∫ 0 1 ( q + v ) g ( v ) 2 d v + O ( q d 2 B 2 e − c κ q ) . \sum_{e^q<q_p\le e^{q+1}}\frac{(\log q_p)^2}{q_p}g(\log q_p-q)^2=\int_0^1(q+v)g(v)^2\,dv+O(qd^2B^2e^{-c\kappa\sqrt{q}}). e q < q p ≤ e q + 1 ∑ q p ( log q p ) 2 g ( log q p − q ) 2 = ∫ 0 1 ( q + v ) g ( v ) 2 d v + O ( q d 2 B 2 e − c κ q ) . The integral is at least q λ 0 B 2 q\lambda_0B^2 q λ 0 B 2 D p D_p D p g + e g+e g + e ∣ e ∣ ≪ d B e − q |e|\ll dBe^{-q} ∣ e ∣ ≪ d B e − q ( g + e ) 2 ≥ g 2 / 2 − e 2 (g+e)^2\ge g^2/2-e^2 ( g + e ) 2 ≥ g 2 /2 − e 2 O ( q d 2 e − 2 q B 2 ) O(qd^2e^{-2q}B^2) O ( q d 2 e − 2 q B 2 ) q 0 q_0 q 0 ∑ D p 2 ≥ c λ 0 e q / ( d 2 ( q + 1 ) ) \sum D_p^2\ge c\lambda_0e^q/(d^2(q+1)) ∑ D p 2 ≥ c λ 0 e q / ( d 2 ( q + 1 )) ∣ ξ j ∣ |\xi_j| ∣ ξ j ∣ e q ≥ c ˉ 0 − 1 d ( q + 1 ) ε 0 ∥ ξ ∥ ∞ e^q\ge\bar c_0^{-1}d(q+1)\varepsilon_0\|\xi\|_\infty e q ≥ c ˉ 0 − 1 d ( q + 1 ) ε 0 ∥ ξ ∥ ∞ exp ( − c λ 0 ε 0 ∥ ξ ∥ ∞ / d ) \exp(-c\lambda_0\varepsilon_0\|\xi\|_\infty/d) exp ( − c λ 0 ε 0 ∥ ξ ∥ ∞ / d )
( 2 e q 0 / ε 0 ) d + 2 d d ! ( d c λ 0 ε 0 ) d ≤ exp ( C d 5 L 0 2 ) . (2e^{q_0}/\varepsilon_0)^d+2^d d!\left(\frac{d}{c\lambda_0\varepsilon_0}\right)^d\le\exp(Cd^5L_0^2). ( 2 e q 0 / ε 0 ) d + 2 d d ! ( c λ 0 ε 0 d ) d ≤ exp ( C d 5 L 0 2 ) . Fourier inversion proves (14) .
Next condition on q p ≤ e Q q_p\le e^Q q p ≤ e Q
δ j 2 ∫ e Q ∞ ( log u ) u − 1 − 2 δ j d u = 1 4 ( 1 + 2 δ j Q ) e − 2 δ j Q ≥ e − 2 / 4 , \delta_j^2\int_{e^Q}^{\infty}(\log u)u^{-1-2\delta_j}\,du=\frac14(1+2\delta_jQ)e^{-2\delta_jQ}\ge e^{-2}/4, δ j 2 ∫ e Q ∞ ( log u ) u − 1 − 2 δ j d u = 4 1 ( 1 + 2 δ j Q ) e − 2 δ j Q ≥ e − 2 /4 , up to O ( e − c κ Q ) O(e^{-c\kappa\sqrt{Q}}) O ( e − c κ Q ) δ j Q ≤ 1 \delta_jQ\le1 δ j Q ≤ 1 θ p a q ( δ j ) a q ( δ k ) + θ p ( 1 − θ p ) c q ( δ j ) c q ( δ k ) \theta_pa_q(\delta_j)a_q(\delta_k)+\theta_p(1-\theta_p)c_q(\delta_j)c_q(\delta_k) θ p a q ( δ j ) a q ( δ k ) + θ p ( 1 − θ p ) c q ( δ j ) c q ( δ k ) O ( e − ∣ t j − t k ∣ ) O(e^{-|t_j-t_k|}) O ( e − ∣ t j − t k ∣ ) h 0 h_0 h 0 c I cI c I 2 ∣ z j z k ∣ ≤ z j 2 + z k 2 2|z_jz_k|\le z_j^2+z_k^2 2∣ z j z k ∣ ≤ z j 2 + z k 2 ℓ 1 \ell^1 ℓ 1
≤ C d 3 ∑ q p > e Q ( log q p ) 3 q p − 3 / 2 ≤ C d 3 ( Q + 1 ) 2 e − Q / 2 . \le Cd^3\sum_{q_p>e^Q}(\log q_p)^3q_p^{-3/2}\le Cd^3(Q+1)^2e^{-Q/2}. ≤ C d 3 q p > e Q ∑ ( log q p ) 3 q p − 3/2 ≤ C d 3 ( Q + 1 ) 2 e − Q /2 . Apply the Taylor replacement argument to ∏ j ψ ( e n j z j ) \prod_j\psi(e^{n_j}z_j) ∏ j ψ ( e n j z j ) ψ = 1 \psi=1 ψ = 1 [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] [ − 2 , 2 ] [-2,2] [ − 2 , 2 ] O ( e 3 N ) O(e^{3N}) O ( e 3 N ) C d C^d C d 4 d e − ∑ n j 4^de^{-\sum n_j} 4 d e − ∑ n j (15) .
Finally put V = ∑ j log + ∣ G X ( t j ) ∣ − 1 V=\sum_j\log^+|G_X(t_j)|^{-1} V = ∑ j log + ∣ G X ( t j ) ∣ − 1 R = d + v R=d+v R = d + v v ≥ 0 v\ge0 v ≥ 0 N = ⌈ B R ⌉ N=\lceil BR\rceil N = ⌈ B R ⌉ Q = D R Q=DR Q = D R δ j > 1 / Q \delta_j>1/Q δ j > 1/ Q δ j ≤ 1 / Q \delta_j\le1/Q δ j ≤ 1/ Q k = O ( log R ) k=O(\log R) k = O ( log R ) N + k N+k N + k N N N (14) , with ε 0 = 1 / Q \varepsilon_0=1/Q ε 0 = 1/ Q
P ( first loss > N + k ) ≤ exp { − B R + C B , D ( 1 + log R ) 7 } . \mathbb{P}(\text{first loss}>N+k)\le\exp\{-BR+C_{B,D}(1+\log R)^7\}. P ( first loss > N + k ) ≤ exp { − B R + C B , D ( 1 + log R ) 7 } . Indeed there are at most ( N + k ) k (N+k)^k ( N + k ) k 2 k e − N 2^ke^{-N} 2 k e − N O ( k 5 log 2 ( 2 k Q ) ) O(k^5\log^2(2kQ)) O ( k 5 log 2 ( 2 k Q )) d 2 d_2 d 2 N + d 2 N+d_2 N + d 2 N N N ( N + d d ) ≤ e E B R \binom{N+d}{d}\le e^{E_BR} ( d N + d ) ≤ e E B R E B = 1 + log ( B + 3 ) E_B=1+\log(B+3) E B = 1 + log ( B + 3 ) (15) bound its probability by
e E B R { C 1 d e − N + C d 3 ( Q + 1 ) 2 e − Q / 2 + 3 N } . e^{E_BR}\{C_1^de^{-N}+Cd^3(Q+1)^2e^{-Q/2+3N}\}. e E B R { C 1 d e − N + C d 3 ( Q + 1 ) 2 e − Q /2 + 3 N } . Choose first B > E B + log C 1 + 4 B>E_B+\log C_1+4 B > E B + log C 1 + 4 D / 2 > E B + 3 ( B + 1 ) + 5 D/2>E_B+3(B+1)+5 D /2 > E B + 3 ( B + 1 ) + 5 D ≥ Q 0 D\ge Q_0 D ≥ Q 0 V > 2 N + d V>2N+d V > 2 N + d P ( V > C 4 ( d + v ) ) ≤ C 5 e − c 4 ( d + v ) \mathbb{P}(V>C_4(d+v))\le C_5e^{-c_4(d+v)} P ( V > C 4 ( d + v )) ≤ C 5 e − c 4 ( d + v ) R R R 0 < a 0 < c 4 / ( 2 C 4 ) 0<a_0<c_4/(2C_4) 0 < a 0 < c 4 / ( 2 C 4 ) (13) .
Corollary 2.8 (One-point negative moment with fixed omissions). Let K K K (5) , and let S S S
P ( X p = ± 1 ) = θ p / 2 , P ( X p = 0 ) = 1 − θ p . \mathbb{P}(X_{\mathfrak{p}}=\pm1)=\theta_{\mathfrak{p}}/2,\qquad\mathbb{P}(X_{\mathfrak{p}}=0)=1-\theta_{\mathfrak{p}}. P ( X p = ± 1 ) = θ p /2 , P ( X p = 0 ) = 1 − θ p . Suppose 0 ≤ θ p ≤ 1 0\leq\theta_{\mathfrak{p}}\leq1 0 ≤ θ p ≤ 1 p \mathfrak{p} p ∣ θ p − θ ∣ ≤ A ∗ ( N p ) − ε ∗ |\theta_{\mathfrak{p}}-\theta|\leq A_*(N\mathfrak{p})^{-\varepsilon_*} ∣ θ p − θ ∣ ≤ A ∗ ( N p ) − ε ∗ S S S 0 < θ ∗ ≤ θ ≤ 1 0<\theta_*\leq\theta\leq1 0 < θ ∗ ≤ θ ≤ 1 G X G_X G X a 0 > 0 a_0>0 a 0 > 0 C < ∞ C<\infty C < ∞
sup t ≥ log 2 E ∣ G X ( t ) ∣ − a 0 ≤ C . (16) \sup_{t\geq\log2}\mathbb{E}|G_X(t)|^{-a_0}\leq C. \tag*{(16)} t ≥ l o g 2 sup E ∣ G X ( t ) ∣ − a 0 ≤ C . (16) In particular one may set θ p = 0 \theta_{\mathfrak{p}}=0 θ p = 0 S S S Q ( i ) \mathbb{Q}(i) Q ( i )
Proof. Enlarge the class constant to
A ∗ ′ = max ( A ∗ , max p ∈ S ( N p ) ε ∗ ) , A_*'=\max\left(A_*,\max_{\mathfrak{p}\in S}(N\mathfrak{p})^{\varepsilon_*}\right), A ∗ ′ = max ( A ∗ , p ∈ S max ( N p ) ε ∗ ) , with the second maximum omitted when S S S ∣ θ p − θ ∣ ≤ 1 |\theta_{\mathfrak{p}}-\theta|\leq1 ∣ θ p − θ ∣ ≤ 1 (6) with A ∗ ′ A_*' A ∗ ′ d = 1 d=1 d = 1
∣ G X ( t ) ∣ − a 0 ≤ exp { a 0 log + ∣ G X ( t ) ∣ − 1 } , |G_X(t)|^{-a_0}\leq\exp\{a_0\log^+|G_X(t)|^{-1}\}, ∣ G X ( t ) ∣ − a 0 ≤ exp { a 0 log + ∣ G X ( t ) ∣ − 1 } , which proves (16) . To make explicit why finite omissions cause no hidden rational-prime reduction, the proof of that lemma uses the prime-ideal counting measure ∑ p ( log N p ) δ N p \sum_{\mathfrak{p}}(\log N\mathfrak{p})\delta_{N\mathfrak{p}} ∑ p ( log N p ) δ N p S S S
Lemma 2.9 (Jensen bounds and exponential zero moments). Let N X m u l t ( u , T ) N_X^{\mathrm{mult}}(u,T) N X mult ( u , T ) G X G_X G X [ u , u + T ] [u,u+T] [ u , u + T ] a , b > 0 a,b>0 a , b > 0
E e a N X m u l t ( u , T ) ≤ e b ( T + 1 ) ( u ≥ log 2 , T ≥ 1 ) . \mathbb{E}e^{aN_X^{\mathrm{mult}}(u,T)}\leq e^{b(T+1)}\qquad(u\geq\log2,\ T\geq1). E e a N X mult ( u , T ) ≤ e b ( T + 1 ) ( u ≥ log 2 , T ≥ 1 ) . The constants are uniform over (6) .
Proof. For a holomorphic f ≢ 0 f\not\equiv0 f ≡ 0 m m m M M M 1 / 8 1/8 1/8 3 / 4 3/4 3/4 t t t z z z m > 0 m>0 m > 0 [ t − 1 / 4 , t + 1 / 4 ] [t-1/4,t+1/4] [ t − 1/4 , t + 1/4 ] ∣ w − z ∣ < 1 / 2 |w-z|<1/2 ∣ w − z ∣ < 1/2 ∣ w − z ∣ ≤ 5 / 8 |w-z|\leq5/8 ∣ w − z ∣ ≤ 5/8 5 / 8 5/8 5/8
N f ( [ t − 1 / 4 , t + 1 / 4 ] ) ≤ log ( M / m ) log ( 5 / 4 ) . (17) N_f([t-1/4,t+1/4])\leq\frac{\log(M/m)}{\log(5/4)}. \tag*{(17)} N f ([ t − 1/4 , t + 1/4 ]) ≤ log ( 5/4 ) log ( M / m ) . (17) This includes endpoints and all multiplicities. For f = G X f=G_X f = G X A ( t ) + log + ∣ G X ( t ) ∣ − 1 A(t)+\log^+|G_X(t)|^{-1} A ( t ) + log + ∣ G X ( t ) ∣ − 1 t j = u + j / 2 t_j=u+j/2 t j = u + j /2 0 ≤ j ≤ ⌈ 2 T ⌉ 0\leq j\leq\lceil2T\rceil 0 ≤ j ≤ ⌈ 2 T ⌉ L L L L / 2 ≥ h 0 L/2\geq h_0 L /2 ≥ h 0 (13) give
E exp { a 0 L ∑ j log + ∣ G X ( t j ) ∣ − 1 } ≤ C 2 T + 2 . \mathbb{E}\exp\left\{\frac{a_0}{L}\sum_j\log^+|G_X(t_j)|^{-1}\right\}\leq C^{2T+2}. E exp { L a 0 j ∑ log + ∣ G X ( t j ) ∣ − 1 } ≤ C 2 T + 2 . Lemma 2.6 controls the sum of the A ( t j ) A(t_j) A ( t j ) a a a
The strong random zero theorem
Theorem 2.10. For the symmetric Euler class over the rational primes, set
c θ = e − θ / 2 π 2 . c_\theta= \frac{e^{-\theta/2}}{\pi\sqrt{2}}. c θ = π 2 e − θ /2 . For every deterministic u ≥ log 2 u \ge\log2 u ≥ log 2 T ≥ 1 T \ge1 T ≥ 1 ⋄ ∈ { dist , mult } \diamond\in\{\operatorname{dist},\operatorname{mult}\} ⋄ ∈ { dist , mult }
∣ E N X ⋄ ( u , T ) − c θ T ∣ ≤ C e − 5 u / 56 . (18) \left|\mathbb{E}N_X^\diamond(u,T)-c_\theta T\right| \le C e^{-5u/56}. \tag*{(18)} ∣ E N X ⋄ ( u , T ) − c θ T ∣ ≤ C e − 5 u /56 . (18) For every η > 0 \eta> 0 η > 0 c η , C η c_\eta,C_\eta c η , C η
P ( ∣ N X ⋄ ( u , T ) − c θ T ∣ > η T ) ≤ C η e − c η T . (19) \mathbb{P}\left(\left|N_X^\diamond(u,T)-c_\theta T\right|>\eta T\right)\le C_\eta e^{-c_\eta T}. \tag*{(19)} P ( ∣ N X ⋄ ( u , T ) − c θ T ∣ > η T ) ≤ C η e − c η T . (19) Moreover the limit
B X ⋄ ( u ) = lim T → ∞ { E N X ⋄ ( u , T ) − c θ T } B_X^\diamond(u)=\lim_{T\to\infty}\left\{\mathbb{E}N_X^\diamond(u,T)-c_\theta T\right\} B X ⋄ ( u ) = T → ∞ lim { E N X ⋄ ( u , T ) − c θ T } exists, has absolute value at most C e − 5 u / 56 Ce^{-5u/56} C e − 5 u /56 O ( e − 5 ( u + T ) / 56 ) O(e^{-5(u+T)/56}) O ( e − 5 ( u + T ) /56 ) B X ⋄ B_X^\diamond B X ⋄
Proof of the expectation estimate. We record both the Gaussian crossing calculation and the control of multiple zeros, since sign changes alone would not prove (18) . Let p θ ( h ) p_\theta(h) p θ ( h ) G ( 0 ) , G ( h ) G(0),G(h) G ( 0 ) , G ( h ) m = v = θ m=v=\theta m = v = θ
U = ( g ( 0 ) + g ( h ) ) / 2 , V = ( g ( h ) − g ( 0 ) ) / h . U=(g(0)+g(h))/2,\qquad V=(g(h)-g(0))/h. U = ( g ( 0 ) + g ( h )) /2 , V = ( g ( h ) − g ( 0 )) / h . They are independent centered Gaussians, with variances ( 1 + r ( h ) ) / 2 = 1 + O ( h 2 ) (1+r(h))/2=1+O(h^2) ( 1 + r ( h )) /2 = 1 + O ( h 2 ) 2 ( 1 − r ( h ) ) / h 2 = 1 / 2 + O ( h 2 ) 2(1-r(h))/h^2=1/2+O(h^2) 2 ( 1 − r ( h )) / h 2 = 1/2 + O ( h 2 ) ∣ U + θ ∣ < h ∣ V ∣ / 2 |U+\sqrt{\theta}|<h|V|/2 ∣ U + θ ∣ < h ∣ V ∣/2 q σ q_\sigma q σ σ \sigma σ
p θ ( h ) h = E [ ∣ V ∣ ∫ − 1 / 2 1 / 2 q ( 1 + r ( h ) ) / 2 ( − θ + h ∣ V ∣ z ) d z ] = c θ + O ( h 2 ) , 0 < h ≤ 1 / 2. \frac{p_\theta(h)}{h}=\mathbb{E}\left[|V|\int_{-1/2}^{1/2}q_{(1+r(h))/2}\left(-\sqrt{\theta}+h|V|z\right)\,dz\right]=c_\theta+O(h^2),\qquad0<h\le1/2. h p θ ( h ) = E [ ∣ V ∣ ∫ − 1/2 1/2 q ( 1 + r ( h )) /2 ( − θ + h ∣ V ∣ z ) d z ] = c θ + O ( h 2 ) , 0 < h ≤ 1/2. The linear Taylor term integrates to zero, the second derivative of the density is uniformly bounded, and E ∣ V ∣ 3 = O ( 1 ) \mathbb{E}|V|^3=O(1) E ∣ V ∣ 3 = O ( 1 ) ( 2 π ) − 1 / 2 e − θ / 2 / π = c θ (2\pi)^{-1/2}e^{-\theta/2}/\sqrt{\pi}=c_\theta ( 2 π ) − 1/2 e − θ /2 / π = c θ θ ∈ [ θ ∗ , 1 ] \theta\in[\theta_*,1] θ ∈ [ θ ∗ , 1 ] (11) gives
p m , v ( h ) = c ( m , v ) h + O ( h 3 ) , c ( m , v ) = 1 π 2 exp ( − m 2 2 v ) . (20) p_{m,v}(h)=c(m,v)h+O(h^3),\qquad c(m,v)=\frac{1}{\pi\sqrt{2}}\exp\left(-\frac{m^2}{2v}\right). \tag*{(20)} p m , v ( h ) = c ( m , v ) h + O ( h 3 ) , c ( m , v ) = π 2 1 exp ( − 2 v m 2 ) . (20) For one or two Euler coordinates at times at least t t t ϵ = e − t / 2 \epsilon=e^{-t/2} ϵ = e − t /2 k k k O ( ϵ − k ) O(\epsilon^{-k}) O ( ϵ − k )
P ( ∣ G X ( v ) ∣ ≤ a ) ≤ C a + C e − v / 2 , P ( G X ( v ) = 0 ) ≤ C e − v / 2 , (21) \mathbb{P}(|G_X(v)|\le a)\le Ca+Ce^{-v/2},\qquad\mathbb{P}(G_X(v)=0)\le Ce^{-v/2}, \tag*{(21)} P ( ∣ G X ( v ) ∣ ≤ a ) ≤ C a + C e − v /2 , P ( G X ( v ) = 0 ) ≤ C e − v /2 , (21) ∣ P ( G X ( v ) G X ( v + h ) < 0 ) − p θ ( h ) ∣ ≤ C e − v / 2 . (22) \left|\mathbb{P}(G_X(v)G_X(v+h)<0)-p_\theta(h)\right|\le Ce^{-v/2}. \tag*{(22)} ∣ P ( G X ( v ) G X ( v + h ) < 0 ) − p θ ( h ) ∣ ≤ C e − v /2 . (22) For example, the smooth crossing test is a ϵ ( x ) ( 1 − a ϵ ( y ) ) + ( 1 − a ϵ ( x ) ) a ϵ ( y ) a_\epsilon(x)(1-a_\epsilon(y))+(1-a_\epsilon(x))a_\epsilon(y) a ϵ ( x ) ( 1 − a ϵ ( y )) + ( 1 − a ϵ ( x )) a ϵ ( y ) a ϵ a_\epsilon a ϵ − ϵ -\epsilon − ϵ ϵ \epsilon ϵ
Lemma 2.2 and Cauchy’s formula give
sup v ≥ log 2 E sup [ v , v + 1 ] ∣ G X ′ ′ ∣ 8 < ∞ . \sup_{v\ge\log2}\mathbb{E}\sup_{[v,v+1]}|G_X''|^8<\infty. v ≥ l o g 2 sup E [ v , v + 1 ] sup ∣ G X ′′ ∣ 8 < ∞. Lemma 2.9 gives a uniform fourth moment for the multiplicity count on a unit interval. For a cell I = [ v , v + d ] I=[v,v+d] I = [ v , v + d ] 0 < d ≤ 1 / 2 0<d\le1/2 0 < d ≤ 1/2 N I N_I N I D I = sup I ∣ G X ′ ′ ∣ D_I=\sup_I|G_X''| D I = sup I ∣ G X ′′ ∣ K = d − 1 / 4 K=d^{-1/4} K = d − 1/4 N I ≥ 2 N_I\ge2 N I ≥ 2 D I ≤ K D_I\le K D I ≤ K ∣ G X ( v ) ∣ ≤ K d 2 |G_X(v)|\le Kd^2 ∣ G X ( v ) ∣ ≤ K d 2 (21) imply
P ( N I ≥ 2 ) ≤ C ( d 7 / 4 + e − v / 2 ) , E [ N I 1 { N I ≥ 2 } ] ≤ C ( d 21 / 16 + e − 3 v / 8 ) . (23) \mathbb{P}(N_I\ge2)\le C(d^{7/4}+e^{-v/2}),\qquad\mathbb{E}[N_I\mathbf{1}_{\{N_I\ge2\}}]\le C(d^{21/16}+e^{-3v/8}). \tag*{(23)} P ( N I ≥ 2 ) ≤ C ( d 7/4 + e − v /2 ) , E [ N I 1 { N I ≥ 2 } ] ≤ C ( d 21/16 + e − 3 v /8 ) . (23) The second bound uses Hölder with the fourth zero-count moment. At a deterministic point v v v Z ( v ) Z(v) Z ( v ) E Z ( v ) ≤ C e − 3 v / 8 \mathbb{E}Z(v)\le Ce^{-3v/8} E Z ( v ) ≤ C e − 3 v /8
Consider [ t , t + L ] [t,t+L] [ t , t + L ] 0 < L ≤ 1 0<L\le1 0 < L ≤ 1 h = 1 2 e − 2 t / 7 h=\frac{1}{2}e^{-2t/7} h = 2 1 e − 2 t /7 n = ⌈ L / h ⌉ ≥ 1 n=\lceil L/h\rceil\ge1 n = ⌈ L / h ⌉ ≥ 1 d = L / n d=L/n d = L / n S S S S S S
S ≤ N d i s t ≤ N m u l t ≤ S + ∑ I N I 1 { N I ≥ 2 } + 2 ∑ mesh v Z ( v ) . S\le N^{\mathrm{dist}}\le N^{\mathrm{mult}}\le S+\sum_I N_I\mathbf{1}_{\{N_I\ge2\}}+2\sum_{\text{mesh }v}Z(v). S ≤ N dist ≤ N mult ≤ S + I ∑ N I 1 { N I ≥ 2 } + 2 mesh v ∑ Z ( v ) . Equations (22) , (R.4), and the crossing calculation give
∣ E S − c θ L ∣ ≤ C ( h 2 + h − 1 e − t / 2 ) , E ( N m u l t − S ) ≤ C ( h 5 / 16 + h − 1 e − 3 t / 8 ) . \left|\mathbb{E}S-c_\theta L\right|\le C(h^2+h^{-1}e^{-t/2}),\qquad\mathbb{E}(N^{\mathrm{mult}}-S)\le C(h^{5/16}+h^{-1}e^{-3t/8}). ∣ E S − c θ L ∣ ≤ C ( h 2 + h − 1 e − t /2 ) , E ( N mult − S ) ≤ C ( h 5/16 + h − 1 e − 3 t /8 ) . Here n ≤ 1 + h − 1 n\le1+h^{-1} n ≤ 1 + h − 1 n d 21 / 16 = L d 5 / 16 ≤ h 5 / 16 nd^{21/16}=Ld^{5/16}\le h^{5/16} n d 21/16 = L d 5/16 ≤ h 5/16 5 t / 56 5t/56 5 t /56
∣ E N ∘ ( t , L ) − c θ L ∣ ≤ C e − 5 t / 56 , 0 ≤ L ≤ 1. (24) \left|\mathbb{E}N^\circ(t,L)-c_\theta L\right|\le Ce^{-5t/56},\qquad0\le L\le1. \tag*{(24)} ∣ E N ∘ ( t , L ) − c θ L ∣ ≤ C e − 5 t /56 , 0 ≤ L ≤ 1. (24) For L = 0 L=0 L = 0
Partition a longer closed interval into unit intervals and a last fraction. The sum of their counts overcounts precisely the shared endpoint roots; their expectations form a convergent geometric sum. Summing (24) proves (18) . The same argument applied to [ u + T 1 , u + T 2 ] [u+T_1,u+T_2] [ u + T 1 , u + T 2 ] C e − 5 ( u + T 1 ) / 56 Ce^{-5(u+T_1)/56} C e − 5 ( u + T 1 ) /56 T 2 ≥ T 1 T_2\ge T_1 T 2 ≥ T 1
For the concentration proof we give the late-interval estimates explicitly. They use a fixed mesh, chosen after the required tolerance.
Lemma 2.11 (Crowded cells and small mesh values). Fix τ , ζ > 0 \tau,\zeta>0 τ , ζ > 0 h > 0 h>0 h > 0 h h h T T T v ≥ log 2 + τ T v\ge\log2+\tau T v ≥ log 2 + τ T
P ( ∑ I N I 1 { N I ≥ 2 } > ζ T ) ≤ C e − c T . \mathbb{P}\left(\sum_I N_I\mathbf{1}_{\{N_I\ge2\}}>\zeta T\right)\le Ce^{-cT}. P ( I ∑ N I 1 { N I ≥ 2 } > ζ T ) ≤ C e − c T . For this h h h ϵ > 0 \epsilon>0 ϵ > 0
P ( ∑ mesh x 1 { ∣ G X ( x ) ∣ ≤ ϵ } > ζ T ) ≤ C e − c T . \mathbb{P}\left(\sum_{\text{mesh }x}\mathbf{1}_{\{|G_X(x)|\le\epsilon\}}>\zeta T\right)\le Ce^{-cT}. P ( mesh x ∑ 1 { ∣ G X ( x ) ∣ ≤ ϵ } > ζ T ) ≤ C e − c T . One may decrease h h h ϵ \epsilon ϵ
Proof. Fix H 0 ≥ max ( h 0 , 1 ) H_0\ge\max(h_0,1) H 0 ≥ max ( h 0 , 1 ) 1 / 2 1/2 1/2 c i c_i c i H 0 H_0 H 0
E e a N ∑ i N i ≤ C N k , E e a D ∑ i D i ≤ C D k , D i = sup I i ∣ G X ′ ∣ . (25) \mathbb{E}e^{a_N\sum_i N_i}\le C_N^{k},\qquad\mathbb{E}e^{a_D\sum_i D_i}\le C_D^{k},\qquad D_i=\sup_{I_i}|G_X'|. \tag*{(25)} E e a N ∑ i N i ≤ C N k , E e a D ∑ i D i ≤ C D k , D i = I i sup ∣ G X ′ ∣. (25) Indeed Jensen at the center uses inner radius 1 / 4 1/4 1/4 3 / 4 3/4 3/4 log ( 5 / 4 ) − 1 \log(5/4)^{-1} log ( 5/4 ) − 1 D i ≤ 8 A ( c i ) D_i\le8A(c_i) D i ≤ 8 A ( c i ) a N > 0 a_N>0 a N > 0 (13) . For every fixed 0 < z < e − 1 0<z<e^{-1} 0 < z < e − 1 B B B (15) gives
P ( ∣ G X ( c i ) ∣ ≤ z for all i ∈ S ) ≤ ( C S z ) ∣ S ∣ + C B , z , τ e − B T . (26) \mathbb{P}(|G_X(c_i)|\le z\text{ for all }i\in S)\le(C_Sz)^{|S|}+C_{B,z,\tau}e^{-BT}. \tag*{(26)} P ( ∣ G X ( c i ) ∣ ≤ z for all i ∈ S ) ≤ ( C S z ) ∣ S ∣ + C B , z , τ e − B T . (26) To verify uniformity, put n z = ⌊ log ( 1 / z ) ⌋ ≥ 1 n_z=\lfloor\log(1/z)\rfloor\ge1 n z = ⌊ log ( 1/ z )⌋ ≥ 1 Q = D T Q=DT Q = D T N = n z ∣ S ∣ N=n_z|S| N = n z ∣ S ∣ T T T Q e − c i ≤ D T e − τ T / 2 ≤ 1 Qe^{-c_i}\le DTe^{-\tau T}/2\le1 Q e − c i ≤ D T e − τ T /2 ≤ 1 ∣ S ∣ ≤ T + 2 |S|\le T+2 ∣ S ∣ ≤ T + 2 (15) is at most C ( T + 2 ) 3 ( D T + 1 ) 2 e − D T / 2 + 3 n z ( T + 2 ) C(T+2)^3(DT+1)^2e^{-DT/2+3n_z(T+2)} C ( T + 2 ) 3 ( D T + 1 ) 2 e − D T /2 + 3 n z ( T + 2 ) D / 2 > B + 3 n z + 2 D/2>B+3n_z+2 D /2 > B + 3 n z + 2 T T T e − n z ≤ e z e^{-n_z}\le ez e − n z ≤ ez
For crowded cells choose K = h − 1 / 4 K=h^{-1/4} K = h − 1/4 z = h 7 / 4 z=h^{7/4} z = h 7/4 { N i ≥ 2 } ⊂ { ∣ G X ( c i ) ∣ ≤ z } ∪ { D i > K } \{N_i\ge2\}\subset\{|G_X(c_i)|\le z\}\cup\{D_i>K\} { N i ≥ 2 } ⊂ { ∣ G X ( c i ) ∣ ≤ z } ∪ { D i > K } k k k 8 / 7 8/7 8/7 8 8 8 (26) and (R.5), the probability that all cells are crowded is at most
( p 0 7 / 8 + p D 7 / 8 ) k + ϵ T 7 / 8 ( 1 + p D 1 / 8 ) k , p 0 = C S h 7 / 4 , p D = C D e − h − 1 / 4 , ϵ T = C B e − B T . (27) \left(p_0^{7/8}+p_D^{7/8}\right)^k+\epsilon_T^{7/8}(1+p_D^{1/8})^k,\qquad p_0=C_Sh^{7/4},\quad p_D=C_De^{-h^{-1/4}},\quad\epsilon_T=C_Be^{-BT}. \tag*{(27)} ( p 0 7/8 + p D 7/8 ) k + ϵ T 7/8 ( 1 + p D 1/8 ) k , p 0 = C S h 7/4 , p D = C D e − h − 1/4 , ϵ T = C B e − B T . (27) The two displayed contributions are a sum, including for an empty subset. Because B B B p ( h ) k + C B e − B T p(h)^k+C_Be^{-BT} p ( h ) k + C B e − B T
p ( h ) = O ( h 49 / 32 ) + O ( e − h − 1 / 4 / 8 ) . p(h)=O(h^{49/32})+O(e^{-h^{-1/4}/8}). p ( h ) = O ( h 49/32 ) + O ( e − h − 1/4 /8 ) . Hölder once more, now with exponents 4 4 4 4 / 3 4/3 4/3
E [ e ( a N / 4 ) ∑ i Y i 1 { N i ≥ 2 ∀ i } ] ≤ q ( h ) k + C N k / 4 C B 3 / 4 e − 3 B T / 4 , \mathbb{E}\left[e^{(a_N/4)\sum_iY_i}\mathbf{1}_{\{N_i\ge2\ \forall i\}}\right]\le q(h)^k+C_N^{k/4}C_B^{3/4}e^{-3BT/4}, E [ e ( a N /4 ) ∑ i Y i 1 { N i ≥ 2 ∀ i } ] ≤ q ( h ) k + C N k /4 C B 3/4 e − 3 B T /4 , where q ( h ) = C N 1 / 4 p ( h ) 3 / 4 = O ( h 147 / 128 ) + O ( e − 3 h − 1 / 4 / 32 ) = o ( h ) q(h)=C_N^{1/4}p(h)^{3/4}=O(h^{147/128})+O(e^{-3h^{-1/4}/32})=o(h) q ( h ) = C N 1/4 p ( h ) 3/4 = O ( h 147/128 ) + O ( e − 3 h − 1/4 /32 ) = o ( h ) ∑ Y i \sum Y_i ∑ Y i Y i = N i 1 { N i ≥ 2 } Y_i=N_i\mathbf{1}_{\{N_i\ge2\}} Y i = N i 1 { N i ≥ 2 } 1 + ( e a N Y i / 4 − 1 ) 1+(e^{a_NY_i/4}-1) 1 + ( e a N Y i /4 − 1 ) m m m
E e ( a N / 4 ) ∑ i Y i ≤ ( 1 + q ( h ) ) m + C N 3 / 4 C B 3 / 4 e − 3 B T / 4 ( 1 + C N 1 / 4 ) m ≤ 2 e q ( h ) m , \mathbb{E}e^{(a_N/4)\sum_iY_i}\le(1+q(h))^m+C_N^{3/4}C_B^{3/4}e^{-3BT/4}(1+C_N^{1/4})^m\le2e^{q(h)m}, E e ( a N /4 ) ∑ i Y i ≤ ( 1 + q ( h ) ) m + C N 3/4 C B 3/4 e − 3 B T /4 ( 1 + C N 1/4 ) m ≤ 2 e q ( h ) m , by choosing B B B L = ⌈ H 0 / h ⌉ L=\lceil H_0/h\rceil L = ⌈ H 0 / h ⌉ E e a N ∑ Y i / L ≤ 2 e q ( h ) n / L \mathbb{E}e^{a_N\sum Y_i/L}\le2e^{q(h)n/L} E e a N ∑ Y i / L ≤ 2 e q ( h ) n / L n ≤ T / h n\le T/h n ≤ T / h ζ T \zeta T ζ T
[ a N ζ h 4 ( H 0 + 1 ) + q ( h ) H 0 ] T . \left[\frac{a_N\zeta h}{4(H_0+1)}+\frac{q(h)}{H_0}\right]T. [ 4 ( H 0 + 1 ) a N ζ h + H 0 q ( h ) ] T . Choose h h h
For the small mesh indicators J ϵ J_\epsilon J ϵ (26) gives
E e J ϵ / L ≤ 2 exp { ( e − 1 ) C S ϵ ( n + 1 ) / L } . \mathbb{E}e^{J_\epsilon/L}\le2\exp\{(e-1)C_S\epsilon(n+1)/L\}. E e J ϵ / L ≤ 2 exp {( e − 1 ) C S ϵ ( n + 1 ) / L } . Choose ϵ ≤ ζ h H 0 / [ 2 ( e − 1 ) C S ( H 0 + 1 ) ] \epsilon\le\zeta hH_0/[2(e-1)C_S(H_0+1)] ϵ ≤ ζ h H 0 / [ 2 ( e − 1 ) C S ( H 0 + 1 )] T T T □ \square □
Lemma 2.12 (Concentration of the smooth sign count). Fix h , ϵ , τ , ζ > 0 h,\epsilon,\tau,\zeta>0 h , ϵ , τ , ζ > 0 x j = v + j h x_j=v+jh x j = v + j h 0 ≤ j ≤ n 0\le j\le n 0 ≤ j ≤ n n ≥ 1 n\ge1 n ≥ 1 n h ≤ T nh\le T nh ≤ T v ≥ log 2 + τ T v\ge\log2+\tau T v ≥ log 2 + τ T
S ϵ = ∑ j < n { a ϵ ( G X ( x j ) ) ( 1 − a ϵ ( G X ( x j + 1 ) ) ) + ( 1 − a ϵ ( G X ( x j ) ) ) a ϵ ( G X ( x j + 1 ) ) } . S_\epsilon=\sum_{j<n}\{a_\epsilon(G_X(x_j))(1-a_\epsilon(G_X(x_{j+1})))+(1-a_\epsilon(G_X(x_j)))a_\epsilon(G_X(x_{j+1}))\}. S ϵ = j < n ∑ { a ϵ ( G X ( x j )) ( 1 − a ϵ ( G X ( x j + 1 ))) + ( 1 − a ϵ ( G X ( x j ))) a ϵ ( G X ( x j + 1 ))} . If m ϵ m_\epsilon m ϵ
P ( ∣ S ϵ − m ϵ ∣ > ζ T ) ≤ C e − c T , ∣ m ϵ − c θ n h ∣ ≤ C T ( h 2 + ϵ / h ) . \mathbb{P}(|S_\epsilon-m_\epsilon|>\zeta T)\le Ce^{-cT},\qquad|m_\epsilon-c_\theta nh|\le CT(h^2+\epsilon/h). P ( ∣ S ϵ − m ϵ ∣ > ζ T ) ≤ C e − c T , ∣ m ϵ − c θ nh ∣ ≤ C T ( h 2 + ϵ / h ) . Proof. The Gaussian covariance on this mesh has operator norm at most C / h C/h C / h ∣ r ( ( j − k ) h ) ∣ |r((j-k)h)| ∣ r (( j − k ) h ) ∣ n + 1 n+1 n + 1 C ( n + 1 ) / ϵ 2 C(n+1)/\epsilon^2 C ( n + 1 ) / ϵ 2 C T / ( h 2 ϵ 2 ) CT/(h^2\epsilon^2) C T / ( h 2 ϵ 2 ) Ent γ ( f 2 ) ≤ 2 ∫ ∣ ∇ f ∣ 2 d γ \operatorname{Ent}_{\gamma}(f^2) \le2\int|\nabla f|^2d\gamma Ent γ ( f 2 ) ≤ 2 ∫ ∣∇ f ∣ 2 d γ [21 Theorem 2.5, pp. 18–19] . For an L L L F F F f = e λ F / 2 f=e^{\lambda F/2} f = e λ F /2 Λ ( λ ) = E e λ F \Lambda(\lambda)=\mathbb{E}e^{\lambda F} Λ ( λ ) = E e λ F λ Λ ′ − Λ log Λ ≤ λ 2 L 2 Λ / 2 \lambda\Lambda' - \Lambda\log\Lambda\le\lambda^2L^2\Lambda/2 λ Λ ′ − Λ log Λ ≤ λ 2 L 2 Λ/2 λ − 1 log Λ \lambda^{-1}\log\Lambda λ − 1 log Λ E F \mathbb{E}F E F E e λ ( F − E F ) ≤ e λ 2 L 2 / 2 \mathbb{E}e^{\lambda(F-\mathbb{E}F)} \le e^{\lambda^2L^2/2} E e λ ( F − E F ) ≤ e λ 2 L 2 /2 F F F
P ( ∣ S ϵ , G − m ϵ ∣ > s T ) ≤ 2 e − c s 2 h 2 ϵ 2 T . \mathbb{P}(|S_{\epsilon,G}-m_{\epsilon}|>sT)\le2e^{-cs^2h^2\epsilon^2T}. P ( ∣ S ϵ , G − m ϵ ∣ > s T ) ≤ 2 e − c s 2 h 2 ϵ 2 T . This is also the argument in [21 pp. 20–21] .
To transfer this event, majorize each tail by a smooth function which changes from zero to one over a band of width ζ T / 2 \zeta T/2 ζ T /2 k ≤ 3 k\le3 k ≤ 3 C ζ ϵ − k C_{\zeta}\epsilon^{-k} C ζ ϵ − k d = n + 1 = O ( T / h ) d=n+1=O(T/h) d = n + 1 = O ( T / h ) b ≤ e − τ T / 2 b\le e^{-\tau T}/2 b ≤ e − τ T /2 C ( T + 1 ) 3 e − τ T ≤ C e − τ T / 2 C(T+1)^3e^{-\tau T}\le Ce^{-\tau T/2} C ( T + 1 ) 3 e − τ T ≤ C e − τ T /2 O ( ϵ ) O(\epsilon) O ( ϵ ) (20) , multiplied by n n n
Proof of (19) . We specify the deterministic counting step. For a real analytic function not identically zero, a mesh of n ≥ 1 n\ge1 n ≥ 1 S S S J ϵ J_{\epsilon} J ϵ
S ≤ N d i s t ≤ N m u l t ≤ S + ∑ I N I 1 { N I ≥ 2 } + 2 ∑ m e s h x 1 { G X ( x ) = 0 } . S\le N^{\mathrm{dist}}\le N^{\mathrm{mult}}\le S+\sum_I N_I\mathbf{1}_{\{N_I\ge2\}}+2\sum_{\mathrm{mesh}\ x}\mathbf{1}_{\{G_X(x)=0\}}. S ≤ N dist ≤ N mult ≤ S + I ∑ N I 1 { N I ≥ 2 } + 2 mesh x ∑ 1 { G X ( x ) = 0 } . Every endpoint root of multiplicity at least two is already included in a crowded adjacent cell; a simple endpoint needs at most the displayed indicator correction. Also ∣ S − S ϵ ∣ ≤ 2 J ϵ |S-S_{\epsilon}|\le2J_{\epsilon} ∣ S − S ϵ ∣ ≤ 2 J ϵ
N m u l t ≤ S ϵ + ∑ I Y I + 4 J ϵ , N d i s t ≥ S ϵ − 2 J ϵ . (28) N^{\mathrm{mult}}\le S_{\epsilon}+\sum_IY_I+4J_{\epsilon},\qquad N^{\mathrm{dist}}\ge S_{\epsilon}-2J_{\epsilon}. \tag*{(28)} N mult ≤ S ϵ + I ∑ Y I + 4 J ϵ , N dist ≥ S ϵ − 2 J ϵ . (28) The condition n ≥ 1 n\ge1 n ≥ 1
Let η ′ = min ( η , 1 ) \eta'=\min(\eta,1) η ′ = min ( η , 1 ) ζ = η ′ / 100 \zeta=\eta'/100 ζ = η ′ /100 a , b a,b a , b Lemma 2.9 . Choose τ ≤ min ( 1 / 4 , a ζ / ( 4 b ) , ζ / c max ) \tau\le\min(1/4,a\zeta/(4b),\zeta/c_{\max}) τ ≤ min ( 1/4 , a ζ / ( 4 b ) , ζ / c m a x ) c max = 1 / ( π 2 ) c_{\max}=1/(\pi\sqrt{2}) c m a x = 1/ ( π 2 ) h h h Lemma 2.11 and C h 2 ≤ ζ / 2 Ch^2\le\zeta/2 C h 2 ≤ ζ /2 ϵ \epsilon ϵ C ϵ / h ≤ ζ / 2 C\epsilon/h\le\zeta/2 C ϵ / h ≤ ζ /2 T T T T T T
v = u + τ T , n = ⌊ ( 1 − τ ) T / h ⌋ ≥ 1 , w = v + n h . v=u+\tau T,\qquad n=\lfloor(1-\tau)T/h\rfloor\ge1,\qquad w=v+nh. v = u + τ T , n = ⌊( 1 − τ ) T / h ⌋ ≥ 1 , w = v + nh . The exponential zero moment shows that the front interval [ u , v ] [u,v] [ u , v ] [ w , u + T ] [w,u+T] [ w , u + T ] h h h ζ T \zeta T ζ T C e − c T Ce^{-cT} C e − c T e b e ( b τ − a ζ ) T e^{b e^{(b\tau-a\zeta)T}} e b e ( b τ − a ζ ) T (28) give, outside another exponentially small set,
N m u l t ( [ v , w ] ) ≤ c ϑ n h + 7 ζ T , N d i s t ( [ v , w ] ) ≥ c ϑ n h − 4 ζ T . N^{\mathrm{mult}}([v,w])\le c_{\vartheta}nh+7\zeta T,\qquad N^{\mathrm{dist}}([v,w])\ge c_{\vartheta}nh-4\zeta T. N mult ([ v , w ]) ≤ c ϑ nh + 7 ζ T , N dist ([ v , w ]) ≥ c ϑ nh − 4 ζ T . The full multiplicity count is at most the sum of the three counts, so is at most c ϑ T + 9 ζ T c_{\vartheta}T+9\zeta T c ϑ T + 9 ζ T c ϑ T − 6 ζ T c_{\vartheta}T-6\zeta T c ϑ T − 6 ζ T c max h ≤ ζ T c_{\max}h\le\zeta T c m a x h ≤ ζ T η T \eta T η T C η C_{\eta} C η T T T
A finite-block polynomial test
Lemma 2.13. Let X α ∈ { 0 , ± 1 } X_{\alpha}\in\{0,\pm1\} X α ∈ { 0 , ± 1 } E X α 2 = θ α \mathbb{E}X_{\alpha}^2=\theta_{\alpha} E X α 2 = θ α a α , b α , μ ∈ R r a_{\alpha},b_{\alpha},\mu\in\mathbb{R}^r a α , b α , μ ∈ R r
ξ α = a α X α + b α ( X α 2 − θ α ) , Y = μ + ∑ α ξ α , Σ = Cov Y . \xi_{\alpha}=a_{\alpha}X_{\alpha}+b_{\alpha}(X_{\alpha}^2-\theta_{\alpha}),\qquad Y=\mu+\sum_{\alpha}\xi_{\alpha},\qquad\Sigma=\operatorname{Cov}Y. ξ α = a α X α + b α ( X α 2 − θ α ) , Y = μ + α ∑ ξ α , Σ = Cov Y . Suppose, for fixed c 0 , C 0 > 0 c_0, C_0 > 0 c 0 , C 0 > 0
c 0 I ≤ Σ ≤ C 0 I , ∥ μ ∥ 2 ≤ C 0 r , ∑ α ( a α a α T + b α b α T ) ≤ C 0 I , β α : = ∥ a α ∥ + ∥ b α ∥ ≤ 1. c_0 I \le\Sigma\le C_0 I,\qquad\|\mu\|^2 \le C_0 r,\qquad\sum_\alpha(a_\alpha a_\alpha^{\mathsf T} + b_\alpha b_\alpha^{\mathsf T}) \le C_0 I,\qquad\beta_\alpha:= \|a_\alpha\| + \|b_\alpha\| \le1. c 0 I ≤ Σ ≤ C 0 I , ∥ μ ∥ 2 ≤ C 0 r , α ∑ ( a α a α T + b α b α T ) ≤ C 0 I , β α := ∥ a α ∥ + ∥ b α ∥ ≤ 1. There is a sufficiently small fixed ϵ 0 > 0 \epsilon_0 > 0 ϵ 0 > 0 r r r A = tr Σ + ∥ μ ∥ 2 A = \operatorname{tr}\Sigma+ \|\mu\|^2 A = tr Σ + ∥ μ ∥ 2 m = ⌊ ϵ 0 r ⌋ m = \lfloor\epsilon_0 r \rfloor m = ⌊ ϵ 0 r ⌋
Q ( z ) = ( A − ∥ z ∥ 2 A − 1 ) 2 m , Z ∼ N ( μ , Σ ) , Q(z) = \left(\frac{A-\|z\|^2}{A-1}\right)^{2m},\qquad Z\sim N(\mu,\Sigma), Q ( z ) = ( A − 1 A − ∥ z ∥ 2 ) 2 m , Z ∼ N ( μ , Σ ) , one has Q ≥ 0 Q \ge0 Q ≥ 0 Q ( z ) ≥ 1 Q(z) \ge1 Q ( z ) ≥ 1 ∥ z ∥ ≤ 1 \|z\| \le1 ∥ z ∥ ≤ 1
E Q ( Z ) ≤ e − c r , ∣ E Q ( Y ) − E Q ( Z ) ∣ ≤ e C r ∑ α β α 3 . (29) \mathbb{E}Q(Z) \le e^{-cr},\qquad\left|\mathbb{E}Q(Y)-\mathbb{E}Q(Z)\right| \le e^{Cr}\sum_\alpha\beta_\alpha^3. \tag*{(29)} E Q ( Z ) ≤ e − cr , ∣ E Q ( Y ) − E Q ( Z ) ∣ ≤ e C r α ∑ β α 3 . (29) The constants depend only on c 0 c_0 c 0 C 0 C_0 C 0 ϵ 0 \epsilon_0 ϵ 0
Proof. We have c 0 r ≤ A ≤ 2 C 0 r c_0r \le A \le2C_0r c 0 r ≤ A ≤ 2 C 0 r r r r A − 1 ≥ c 0 r / 2 A-1 \ge c_0r/2 A − 1 ≥ c 0 r /2 m ≥ ϵ 0 r / 2 ≥ 1 m \ge\epsilon_0r/2 \ge1 m ≥ ϵ 0 r /2 ≥ 1 Z i = u i + λ i g i Z_i = u_i + \sqrt{\lambda_i}g_i Z i = u i + λ i g i c 0 ≤ λ i ≤ C 0 c_0 \le\lambda_i \le C_0 c 0 ≤ λ i ≤ C 0 V = ∥ Z ∥ 2 − A V=\|Z\|^2-A V = ∥ Z ∥ 2 − A
log E e t V = ∑ i { − 1 2 log ( 1 − 2 t λ i ) − t λ i + 2 t 2 λ i u i 2 1 − 2 t λ i } ≤ C t 2 r , ∣ t ∣ ≤ ( 4 C 0 ) − 1 . \log\mathbb{E}e^{tV}=\sum_i\left\{-\frac{1}{2}\log(1-2t\lambda_i)-t\lambda_i+\frac{2t^2\lambda_i u_i^2}{1-2t\lambda_i}\right\}\le Ct^2r,\qquad|t|\le(4C_0)^{-1}. log E e t V = i ∑ { − 2 1 log ( 1 − 2 t λ i ) − t λ i + 1 − 2 t λ i 2 t 2 λ i u i 2 } ≤ C t 2 r , ∣ t ∣ ≤ ( 4 C 0 ) − 1 . Chernoff with both signs, choosing t t t min ( v / r , 1 ) \min(v/r,1) min ( v / r , 1 )
P ( ∣ V ∣ > v ) ≤ 2 e − c min ( v 2 / r , v ) . \mathbb{P}(|V|>v)\le2e^{-c\min(v^2/r,v)}. P ( ∣ V ∣ > v ) ≤ 2 e − c m i n ( v 2 / r , v ) . Tail integration therefore gives ∥ V ∥ q ≤ C ( r q + q ) \|V\|_q\le C(\sqrt{rq}+q) ∥ V ∥ q ≤ C ( r q + q ) q ≥ 2 q\ge2 q ≥ 2 q = 2 m q=2m q = 2 m A − 1 A-1 A − 1 C ( ϵ 0 + ϵ 0 ) C(\sqrt{\epsilon_0}+\epsilon_0) C ( ϵ 0 + ϵ 0 ) ϵ 0 \epsilon_0 ϵ 0 1 / 2 1/2 1/2
E Q ( Z ) ≤ 2 − 2 m ≤ e − c r . \mathbb{E}Q(Z)\le2^{-2m}\le e^{-cr}. E Q ( Z ) ≤ 2 − 2 m ≤ e − cr . Replace the ξ α \xi_\alpha ξ α g α g_\alpha g α W W W E e ⟨ h , W ⟩ ≤ e C ∥ h ∥ 2 \mathbb{E}e^{\langle h,W\rangle}\le e^{C\|h\|^2} E e ⟨ h , W ⟩ ≤ e C ∥ h ∥ 2 8 C 0 ∥ h ∥ 2 8C_0\|h\|^2 8 C 0 ∥ h ∥ 2 1 / 2 1/2 1/2 1 / 2 1/2 1/2 5 r 5^r 5 r 1 / 4 1/4 1/4 P ( ∥ W ∥ > C r + s ) ≤ 2 e − c s 2 \mathbb{P}(\|W\|>C\sqrt r+s)\le2e^{-cs^2} P ( ∥ W ∥ > C r + s ) ≤ 2 e − c s 2
( E ∥ W + μ ∥ q ) 1 / q ≤ C ( r + q ) , q ≥ 2. (30) \left(\mathbb{E}\|W+\mu\|^q\right)^{1/q}\le C(\sqrt r+\sqrt q),\qquad q\ge2. \tag*{(30)} ( E ∥ W + μ ∥ q ) 1/ q ≤ C ( r + q ) , q ≥ 2. (30) Write p = 2 m p=2m p = 2 m q ( z ) = A − ∥ z ∥ 2 q(z)=A-\|z\|^2 q ( z ) = A − ∥ z ∥ 2
D 3 Q ( z ) [ h , h , h ] = ( A − 1 ) − p { p ( p − 1 ) ( p − 2 ) q p − 3 ( D q [ h ] ) 3 + 3 p ( p − 1 ) q p − 2 D q [ h ] D 2 q [ h , h ] } . D^3Q(z)[h,h,h]=(A-1)^{-p}\{p(p-1)(p-2)q^{p-3}(Dq[h])^3+3p(p-1)q^{p-2}Dq[h]D^2q[h,h]\}. D 3 Q ( z ) [ h , h , h ] = ( A − 1 ) − p { p ( p − 1 ) ( p − 2 ) q p − 3 ( D q [ h ] ) 3 + 3 p ( p − 1 ) q p − 2 D q [ h ] D 2 q [ h , h ]} . When p = 2 p=2 p = 2 q = 0 q=0 q = 0 ∣ D q [ h ] ∣ ≤ 2 ∥ z ∥ ∥ h ∥ |Dq[h]|\le2\|z\|\|h\| ∣ D q [ h ] ∣ ≤ 2∥ z ∥∥ h ∥ ∣ D 2 q [ h , h ] ∣ ≤ 2 ∥ h ∥ 2 |D^2q[h,h]|\le2\|h\|^2 ∣ D 2 q [ h , h ] ∣ ≤ 2∥ h ∥ 2 p = O ( r ) p=O(r) p = O ( r ) A − 1 ≥ c r A-1\ge cr A − 1 ≥ cr
∣ D 3 Q ( z ) [ h , h , h ] ∣ ≤ e C r ∥ h ∥ 3 ( 1 + ∥ z ∥ / r ) 2 p + 3 . (31) |D^3Q(z)[h,h,h]|\le e^{Cr}\|h\|^3(1+\|z\|/\sqrt r)^{2p+3}. \tag*{(31)} ∣ D 3 Q ( z ) [ h , h , h ] ∣ ≤ e C r ∥ h ∥ 3 ( 1 + ∥ z ∥/ r ) 2 p + 3 . (31) For z = W + μ + t h z=W+\mu+th z = W + μ + t h 0 ≤ t ≤ 1 0\le t\le1 0 ≤ t ≤ 1 W + μ W+\mu W + μ h h h e C r e^{Cr} e C r (30) , since its power is O ( r ) O(r) O ( r ) h = ξ α h=\xi_\alpha h = ξ α ∥ h ∥ ≤ β α ≤ 1 \|h\|\le\beta_\alpha\le1 ∥ h ∥ ≤ β α ≤ 1
E [ ∥ h ∥ 3 ( 1 + ∥ h ∥ / r ) 2 p + 3 ] ≤ e C r β α 3 . \mathbb{E}[\|h\|^3(1+\|h\|/\sqrt r)^{2p+3}]\le e^{Cr}\beta_\alpha^3. E [ ∥ h ∥ 3 ( 1 + ∥ h ∥/ r ) 2 p + 3 ] ≤ e C r β α 3 . For h = g α h=g_\alpha h = g α β α 2 \beta_\alpha^2 β α 2 E ∥ g α ∥ k ≤ ( C k β α ) k \mathbb{E}\lVert g_\alpha\rVert^k \le(C\sqrt{k}\,\beta_\alpha)^k E ∥ g α ∥ k ≤ ( C k β α ) k K = 2 p + 3 K=2p+3 K = 2 p + 3
∑ j = 0 K ( K j ) r − j / 2 E ∥ g α ∥ j + 3 ≤ β α 3 ∑ j = 0 K ( K j ) C j + 3 ( j + 3 ) ( j + 3 ) / 2 r − j / 2 ≤ e C r β α 3 . \sum_{j=0}^{K}\binom{K}{j}r^{-j/2}\mathbb{E}\lVert g_\alpha\rVert^{j+3}\le\beta_\alpha^3\sum_{j=0}^{K}\binom{K}{j}C^{j+3}(j+3)^{(j+3)/2}r^{-j/2}\le e^{C r}\beta_\alpha^3. j = 0 ∑ K ( j K ) r − j /2 E ∥ g α ∥ j + 3 ≤ β α 3 j = 0 ∑ K ( j K ) C j + 3 ( j + 3 ) ( j + 3 ) /2 r − j /2 ≤ e C r β α 3 . The omitted block is independent of W W W e C r β α 3 e^{C r}\beta_\alpha^3 e C r β α 3 (29) . □
Finite Euler polynomials and finite-word orthogonality
We use the explicit weight
w y ( n ) = log + 2 ( y / n ) − 2 log + 2 ( y 2 / n ) + log + 2 ( y 3 / n ) 2 ( log y ) 2 , y = e ℓ , ℓ ≥ 8. (32) w_y(n)=\frac{\log_+^2(y/n)-2\log_+^2(y^2/n)+\log_+^2(y^3/n)}{2(\log y)^2},\qquad y=e^\ell,\quad\ell\ge8. \tag*{(32)} w y ( n ) = 2 ( log y ) 2 log + 2 ( y / n ) − 2 log + 2 ( y 2 / n ) + log + 2 ( y 3 / n ) , y = e ℓ , ℓ ≥ 8. (32) It is between zero and one, equals one for n ≤ y n\le y n ≤ y n > y 3 n>y^3 n > y 3 log n = ℓ , 2 ℓ , 3 ℓ \log n=\ell,2\ell,3\ell log n = ℓ , 2 ℓ , 3 ℓ (8) put
P y , E ( s ) = ∑ p k ≤ y 3 w y ( p k ) ( log p ) a p , k p − k s . P_{y,E}(s)=\sum_{p^k\le y^3}w_y(p^k)(\log p)a_{p,k}p^{-ks}. P y , E ( s ) = p k ≤ y 3 ∑ w y ( p k ) ( log p ) a p , k p − k s . An arithmetic family F X \mathcal{F}_X F X a p , k ( f ) a_{p,k}(f) a p , k ( f ) W = ∏ j = 1 r a p j , k j W=\prod_{j=1}^{r}a_{p_j,k_j} W = ∏ j = 1 r a p j , k j n ( W ) = ∏ j p j k j n(W)=\prod_j p_j^{k_j} n ( W ) = ∏ j p j k j
∣ E X W − E E W ∣ ≤ C X − κ C 0 r n ( W ) A , κ > 0 , A ≥ 0 , C 0 ≥ 1. (33) \left|\mathbb{E}_XW-\mathbb{E}_EW\right|\le CX^{-\kappa}C_0^{r}n(W)^A,\qquad\kappa>0,\quad A\ge0,\quad C_0\ge1. \tag*{(33)} ∣ E X W − E E W ∣ ≤ C X − κ C 0 r n ( W ) A , κ > 0 , A ≥ 0 , C 0 ≥ 1. (33) The outside constant is uniform in every word and X X X
Lemma 2.14 (The arithmetic norm). Fix a formal representation P = ∑ W c W W P=\sum_W c_WW P = ∑ W c W W and set
∥ P ∥ a r = ∑ W ∣ c W ∣ C 0 r ( W ) n ( W ) A . \lVert P\rVert_{\mathrm{ar}}=\sum_W|c_W|C_0^{r(W)}n(W)^A. ∥ P ∥ ar = W ∑ ∣ c W ∣ C 0 r ( W ) n ( W ) A . Products are represented by word concatenation. Then
∥ P Q ∥ a r ≤ ∥ P ∥ a r ∥ Q ∥ a r , ∣ E X P − E E P ∣ ≤ C X − κ ∥ P ∥ a r . \lVert PQ\rVert_{\mathrm{ar}}\le\lVert P\rVert_{\mathrm{ar}}\lVert Q\rVert_{\mathrm{ar}},\qquad\left|\mathbb{E}_XP-\mathbb{E}_EP\right|\le CX^{-\kappa}\lVert P\rVert_{\mathrm{ar}}. ∥ P Q ∥ ar ≤ ∥ P ∥ ar ∥ Q ∥ ar , ∣ E X P − E E P ∣ ≤ C X − κ ∥ P ∥ ar . For ℜ s ≥ 1 / 2 \Re s\ge1/2 ℜ s ≥ 1/2 ∣ a ∣ |a| ∣ a ∣
∥ a P y ( s ) ∥ a r ≤ C e C A ℓ . (34) \lVert aP_y(s)\rVert_{\mathrm{ar}}\le Ce^{C A\ell}. \tag*{(34)} ∥ a P y ( s ) ∥ ar ≤ C e C A ℓ . (34) There is no need to minimize over equivalent formal representations.
Proof. Both r ( W ) r(W) r ( W ) log n ( W ) \log n(W) log n ( W ) (33) proves the expectation bound. Each term in a P y aP_y a P y C 0 C_0 C 0 C log ( p k ) ( p k ) − 1 / 2 C\log(p^k)(p^k)^{-1/2} C log ( p k ) ( p k ) − 1/2 n ≤ e 3 ℓ n\le e^{3\ell} n ≤ e 3 ℓ ∑ n ≤ e 3 ℓ ( log n ) n A − 1 / 2 ≤ C e C A ℓ \sum_{n\le e^{3\ell}}(\log n)n^{A-1/2}\le Ce^{CA\ell} ∑ n ≤ e 3 ℓ ( log n ) n A − 1/2 ≤ C e C A ℓ (34) . □
Lemma 2.15 (Random polynomial moments and tails). In a fixed right sector and for bounded a a a Q E ( a ) = a P y , E ( 1 / 2 + a ) Q_E(a)=aP_{y,E}(1/2+a) Q E ( a ) = a P y , E ( 1/2 + a ) Then, uniformly for q ≥ 2 q\ge2 q ≥ 2
∥ Q E ( a ) ∥ q ≤ C q . \lVert Q_E(a)\rVert_q\le C\sqrt{q}. ∥ Q E ( a ) ∥ q ≤ C q . If ℜ ( a ) ℓ ≥ 1 \Re(a)\ell\ge1 ℜ ( a ) ℓ ≥ 1
∥ a F ( a ) − Q E ( a ) ∥ 2 ≤ C ( 1 + ∣ a ∣ ℓ ) C e − c ℜ ( a ) ℓ . (35) \lVert aF(a)-Q_E(a)\rVert_2\le C(1+|a|\ell)C e^{-c\Re(a)\ell}. \tag*{(35)} ∥ a F ( a ) − Q E ( a ) ∥ 2 ≤ C ( 1 + ∣ a ∣ ℓ ) C e − c ℜ ( a ) ℓ . (35) For real a j = e − t j a_j=e^{-t_j} a j = e − t j t j ≥ log 2 t_j\ge\log2 t j ≥ log 2 let V f = ∑ j τ j a j P y , f ( 1 / 2 + a j ) V_f=\sum_j\tau_j a_jP_{y,f}(1/2+a_j) V f = ∑ j τ j a j P y , f ( 1/2 + a j ) V E = ∑ j τ j Q E ( a j ) V_E=\sum_j\tau_jQ_E(a_j) V E = ∑ j τ j Q E ( a j ) and T = ∑ j ∣ τ j ∣ T=\sum_j|\tau_j| T = ∑ j ∣ τ j ∣ For every integer M ≥ 1 M\ge1 M ≥ 1
E X ∣ V f ∣ 2 M ≤ ( C T M ) 2 M + C X − κ ( C T e C A ℓ ) 2 M . (36) \mathbb{E}_X|V_f|^{2M}\le(CT\sqrt{M})^{2M}+CX^{-\kappa}(CTe^{CA\ell})^{2M}. \tag*{(36)} E X ∣ V f ∣ 2 M ≤ ( C T M ) 2 M + C X − κ ( C T e C A ℓ ) 2 M . (36) This moment bound also holds for complex τ j \tau_j τ j a j a_j a j T T T
Proof. The squared range lengths of the centered prime blocks of Q E ( a ) Q_E(a) Q E ( a ) (10) . The bounded-variable exponential estimate therefore gives L q L^q L q O ( q ) O(\sqrt{q}) O ( q ) c p c_p c p (8) against the nonincreasing weight w y ( p 2 ) w_y(p^2) w y ( p 2 )
∣ a ∑ p w y ( p 2 ) c p ( log p ) p − 1 − 2 a ∣ ≤ C . \left|a\sum_p w_y(p^2)c_p(\log p)p^{-1-2a}\right|\le C. a p ∑ w y ( p 2 ) c p ( log p ) p − 1 − 2 a ≤ C . Explicitly the partial sums of c p log p c_p\log p c p log p O ( x ) O(x) O ( x ) x − 1 − 2 a x^{-1-2a} x − 1 − 2 a O ( 1 / ℜ a ) O(1/\Re a) O ( 1/ℜ a ) ∣ a ∣ |a| ∣ a ∣
For the tail, the centered linear variance is bounded by
C ∣ a ∣ 2 ∑ p > y ( log p ) 2 p − 1 − 2 ℜ a ≤ C ( 1 + ∣ a ∣ ℓ ) e − 2 ℜ ( a ) ℓ . C|a|^2\sum_{p>y}(\log p)^2p^{-1-2\Re a}\le C(1+|a|\ell)e^{-2\Re(a)\ell}. C ∣ a ∣ 2 p > y ∑ ( log p ) 2 p − 1 − 2ℜ a ≤ C ( 1 + ∣ a ∣ ℓ ) e − 2ℜ ( a ) ℓ . The centered degree-two terms are bounded in L 2 L^2 L 2 O ( ( 1 + ∣ a ∣ ℓ ) C e − c ℜ ( a ) ℓ ) O((1+|a|\ell)^C e^{-c\Re(a)\ell}) O (( 1 + ∣ a ∣ ℓ ) C e − c ℜ ( a ) ℓ ) p > y p>\sqrt{y} p > y 1 − w y ( p 2 ) 1-w_y(p^2) 1 − w y ( p 2 ) ℓ \ell ℓ p k > y p^k>y p k > y
( p k ) − 1 / 2 − ℜ a ≤ y − 1 / 12 ( p k ) − 5 / 12 − ℜ a . (p^k)^{-1/2-\Re a}\le y^{-1/12}(p^k)^{-5/12-\Re a}. ( p k ) − 1/2 − ℜ a ≤ y − 1/12 ( p k ) − 5/12 − ℜ a . The sum over p , k ≥ 3 p,k\ge3 p , k ≥ 3 ( log p ) ( p k ) − 5 / 12 − ℜ a (\log p)(p^k)^{-5/12-\Re a} ( log p ) ( p k ) − 5/12 − ℜ a 5 / 4 5/4 5/4 C e − ℓ / 12 Ce^{-\ell/12} C e − ℓ /12 (35) , with a possibly smaller sector constant c c c
Minkowski gives ∥ V E ∥ 2 M ≤ C T M \|V_E\|_{2M}\le CT\sqrt{M} ∥ V E ∥ 2 M ≤ C T M V f V_f V f 2 M 2M 2 M V f 2 M V_f^{2M} V f 2 M (34) to this polynomial to obtain (36) . For complex values apply the identical argument to V f M V f ‾ M V_f^M\overline{V_f}^M V f M V f M a p , k ( f ) a_{p,k}(f) a p , k ( f )
Proposition 2.16 (Characteristic-function transfer). In the notation above, for every integer M ≥ 1 M\ge1 M ≥ 1
∣ E X e i V f − E E e i V E ∣ ≤ 2 ( C T / 2 M ) 2 M + C X − κ exp { C M ( ℓ + log ( 2 + T ) ) } . (37) \left|\mathbb{E}Xe^{iV_f}-\mathbb{E}_Ee^{iV_E}\right|\le2(CT/\sqrt{2M})^{2M}+CX^{-\kappa}\exp\{CM(\ell+\log(2+T))\}. \tag*{(37)} E X e i V f − E E e i V E ≤ 2 ( C T / 2 M ) 2 M + C X − κ exp { C M ( ℓ + log ( 2 + T ))} . (37) This applies, in particular, to every fixed-dimensional vector. The displayed bound also keeps all dimension dependence through T T T Φ \Phi Φ Ω \Omega Ω
∣ E X Φ ( A f ) − E E Φ ( A E ) ∣ ≤ ( 2 π ) − d { ∫ Ω ∣ Φ ^ ( τ ) ∣ Δ ( τ ) d τ + 2 ∫ Ω c ∣ Φ ^ ( τ ) ∣ d τ } , \left|\mathbb{E}_X\Phi(A_f)-\mathbb{E}_E\Phi(A_E)\right|
\le(2\pi)^{-d}\left\{\int_\Omega|\widehat{\Phi}(\tau)|\Delta(\tau)\,d\tau
+2\int_{\Omega^c}|\widehat{\Phi}(\tau)|\,d\tau\right\}, ∣ E X Φ ( A f ) − E E Φ ( A E ) ∣ ≤ ( 2 π ) − d { ∫ Ω ∣ Φ ( τ ) ∣Δ ( τ ) d τ + 2 ∫ Ω c ∣ Φ ( τ ) ∣ d τ } , where Δ ( τ ) \Delta(\tau) Δ ( τ ) (37) . Thus a growing Fourier norm or tail is never discarded.
Proof. Taylor-expand e i V e^{iV} e iV 2 M − 1 2M-1 2 M − 1 V V V ∣ V ∣ 2 M / ( 2 M ) ! |V|^{2M}/(2M)! ∣ V ∣ 2 M / ( 2 M )! ( 2 M ) ! ≥ ( 2 M / e ) 2 M (2M)!\ge(2M/e)^{2M} ( 2 M )! ≥ ( 2 M / e ) 2 M
C X − κ ∑ j < 2 M ∥ V ∥ a r j j ! . CX^{-\kappa}\sum_{j<2M}\frac{\|V\|_{\mathrm{ar}}^j}{j!}. C X − κ j < 2 M ∑ j ! ∥ V ∥ ar j . Using ∥ V ∥ a r ≤ C T e C A ℓ \lVert V\rVert_{\mathrm{ar}} \le CT e^{CA\ell} ∥ V ∥ ar ≤ C T e C A ℓ (37) . No norm of an infinite exponential is used. Fourier inversion and Fubini give the final statement; outside Ω \Omega Ω
Lemma 2.17 (The ideal arithmetic norm). Suppose a f a_f a f
∣ E μ x a f ( a ) − E X ( a ) ∣ ≤ C x − κ 0 ( N a ) λ . \left|\mathbb{E}_{\mu_x}a_f(\mathfrak{a})-\mathbb{E}X(\mathfrak{a})\right| \le Cx^{-\kappa_0}(N\mathfrak{a})^\lambda. ∣ E μ x a f ( a ) − E X ( a ) ∣ ≤ C x − κ 0 ( N a ) λ . For P = ∑ a c a a f ( a ) P=\sum_{\mathfrak{a}}c_{\mathfrak{a}}a_f(\mathfrak{a}) P = ∑ a c a a f ( a ) ∥ P ∥ a r , K = ∑ a ∣ c a ∣ ( N a ) λ \lVert P\rVert_{\mathrm{ar},K}=\sum_{\mathfrak{a}}|c_{\mathfrak{a}}|(N\mathfrak{a})^\lambda ∥ P ∥ ar , K = ∑ a ∣ c a ∣ ( N a ) λ C x − κ 0 Cx^{-\kappa_0} C x − κ 0 P y ( s ) P_y(s) P y ( s ) ∑ q p ≤ y 3 w y ( q p ) a f ( p ) q p − s \sum_{q_{\mathfrak{p}}\le y^3}w_y(q_{\mathfrak{p}})a_f(\mathfrak{p})q_{\mathfrak{p}}^{-s} ∑ q p ≤ y 3 w y ( q p ) a f ( p ) q p − s C e C ℓ Ce^{C\ell} C e C ℓ ℜ s ≥ 1 / 2 \Re s\ge1/2 ℜ s ≥ 1/2 S f = ∑ p c p a f ( p ) S_f=\sum_{\mathfrak{p}}c_{\mathfrak{p}}a_f(\mathfrak{p}) S f = ∑ p c p a f ( p )
E μ x ∣ S f ∣ 2 m ≤ ( C m ) m ( ∑ ∣ c p ∣ 2 ) m + C x − κ 0 ( ∑ ∣ c p ∣ q p λ ) 2 m . \mathbb{E}_{\mu_x}|S_f|^{2m}\le(Cm)^m\left(\sum|c_{\mathfrak{p}}|^2\right)^m+Cx^{-\kappa_0}\left(\sum|c_{\mathfrak{p}}|q_{\mathfrak{p}}^\lambda\right)^{2m}. E μ x ∣ S f ∣ 2 m ≤ ( C m ) m ( ∑ ∣ c p ∣ 2 ) m + C x − κ 0 ( ∑ ∣ c p ∣ q p λ ) 2 m . Proof. Complete multiplicativity makes the product coefficients the convolution over a b = c \mathfrak{a}\mathfrak{b}=\mathfrak{c} ab = c n K = [ K : Q ] n_K=[K:\mathbb{Q}] n K = [ K : Q ] n n n d n K ( n ) d_{n_K}(n) d n K ( n ) ∏ i = 1 r ( 1 − z f i ) − 1 \prod_{i=1}^r(1-z^{f_i})^{-1} ∏ i = 1 r ( 1 − z f i ) − 1 ( 1 − z ) − n K (1-z)^{-n_K} ( 1 − z ) − n K
# { a : N a ≤ Y } ≤ Y ( 1 + log Y ) n K − 1 . \#\{\mathfrak{a}:N\mathfrak{a}\le Y\}\le Y(1+\log Y)^{n_K-1}. # { a : N a ≤ Y } ≤ Y ( 1 + log Y ) n K − 1 . Since the ideal von Mangoldt function is at most log N a \log N\mathfrak{a} log N a ( y 3 ) λ + 1 log ( y 3 ) ( 1 + 3 ℓ ) n K − 1 ≤ C e C ℓ (y^3)^{\lambda+1}\log(y^3)(1+3\ell)^{n_K-1}\le Ce^{C\ell} ( y 3 ) λ + 1 log ( y 3 ) ( 1 + 3 ℓ ) n K − 1 ≤ C e C ℓ 2 m 2m 2 m ( 2 m ) ! / ( 2 m m ! ) (2m)!/(2^m m!) ( 2 m )! / ( 2 m m !) m m m S f m S f ‾ m S_f^m\overline{S_f}^m S f m S f m p q \mathfrak{p}\mathfrak{q} pq N p = N q N\mathfrak{p}=N\mathfrak{q} N p = N q
Lemma 2.18 (Analytic coefficients near an endpoint). For the symmetric ideal model, 0 < Δ ≤ 1 / 2 0<\Delta\le1/2 0 < Δ ≤ 1/2 ∣ z ∣ , ∣ w ∣ ≤ 1 / 2 |z|,|w|\le1/2 ∣ z ∣ , ∣ w ∣ ≤ 1/2
a ( z ) = Δ 1 + z 1 − z , H Δ ( z ) = Δ ( 1 − z ) 2 ( L X ′ L X ) ( 1 / 2 + a ( z ) ) . a(z)=\Delta\frac{1+z}{1-z},\qquad H_\Delta(z)=\frac{\Delta}{(1-z)^2}\left(\frac{L'_X}{L_X}\right)(1/2+a(z)). a ( z ) = Δ 1 − z 1 + z , H Δ ( z ) = ( 1 − z ) 2 Δ ( L X L X ′ ) ( 1/2 + a ( z )) . Then
E H Δ ( z ) = θ 2 ( 1 − z 2 ) + O ( Δ ) , Cov ( H Δ ( z ) , H Δ ( w ) ) = θ 4 ( 1 − z w ) 2 + O ( Δ 2 ) . \mathbb{E}H_\Delta(z)=\frac{\theta}{2(1-z^2)}+O(\Delta),\qquad\operatorname{Cov}(H_\Delta(z),H_\Delta(w))=\frac{\theta}{4(1-zw)^2}+O(\Delta^2). E H Δ ( z ) = 2 ( 1 − z 2 ) θ + O ( Δ ) , Cov ( H Δ ( z ) , H Δ ( w )) = 4 ( 1 − z w ) 2 θ + O ( Δ 2 ) . In particular, if Z k = 2 [ z k ] H Δ ( z ) / θ ( k + 1 ) Z_k=2[z^k]H_\Delta(z)/\sqrt{\theta(k+1)} Z k = 2 [ z k ] H Δ ( z ) / θ ( k + 1 )
Cov ( Z k , Z l ) = 1 k = l + O ( Δ 2 2 k + l ) , ∑ k < r ∣ E Z k ∣ 2 ≤ C ( 1 + log ( r + 1 ) ) + C Δ 2 4 r . \operatorname{Cov}(Z_k,Z_l)=1_{k=l}+O(\Delta^2 2^{k+l}),\qquad\sum_{k<r}|\mathbb{E}Z_k|^2\le C(1+\log(r+1))+C\Delta^2 4^r. Cov ( Z k , Z l ) = 1 k = l + O ( Δ 2 2 k + l ) , k < r ∑ ∣ E Z k ∣ 2 ≤ C ( 1 + log ( r + 1 )) + C Δ 2 4 r . Replacing the logarithmic derivative in H Δ H_\Delta H Δ P y P_y P y L 2 L^2 L 2 ∣ z ∣ ≤ 1 / 2 |z|\le1/2 ∣ z ∣ ≤ 1/2 C e − Δ ℓ / 3 Ce^{-\Delta\ell/3} C e − Δ ℓ /3
Proof. Divide (9) by its normalizing factors. The two exact identities
Δ ( 1 − z ) 2 1 2 a ( z ) = 1 2 ( 1 − z 2 ) , Δ 2 ( 1 − z ) 2 ( 1 − w ) 2 ( a ( z ) + a ( w ) ) 2 = 1 4 ( 1 − z w ) 2 \frac{\Delta}{(1-z)^2}\frac{1}{2a(z)}=\frac{1}{2(1-z^2)},\qquad\frac{\Delta^2}{(1-z)^2(1-w)^2(a(z)+a(w))^2}=\frac{1}{4(1-zw)^2} ( 1 − z ) 2 Δ 2 a ( z ) 1 = 2 ( 1 − z 2 ) 1 , ( 1 − z ) 2 ( 1 − w ) 2 ( a ( z ) + a ( w ) ) 2 Δ 2 = 4 ( 1 − z w ) 2 1 give the first formulas. Cauchy on circles of radius 1 / 2 1/2 1/2 θ / ( k + 1 ) \theta/(k+1) θ / ( k + 1 ) k k k k k k
For the sharper residual tail bound in this symmetric model, E [ X ( a ) X ( b ) ] ≥ 0 \mathbb{E}[X(\mathfrak{a})X(\mathfrak{b})]\ge0 E [ X ( a ) X ( b )] ≥ 0 r z = ℜ a ( z ) ≥ Δ / 3 r_z=\Re a(z)\ge\Delta/3 r z = ℜ a ( z ) ≥ Δ/3
norms exceed y y y 0 ≤ 1 − w y ≤ 1 0 \le1-w_y \le1 0 ≤ 1 − w y ≤ 1
E ∣ L X ( 1 / 2 + a ( z ) ) − P y , X ( 1 / 2 + a ( z ) ) ∣ 2 ≤ y − r z E ∣ L X ( 1 / 2 + r z / 2 ) ∣ 2 ≤ C y − r z r z − 2 . \mathbb{E}\left|L_X(1/2+a(z))-P_{y,X}(1/2+a(z))\right|^2 \le y^{-r_z}\mathbb{E}\left|L_X(1/2+r_z/2)\right|^2 \le Cy^{-r_z}r_z^{-2}. E ∣ L X ( 1/2 + a ( z )) − P y , X ( 1/2 + a ( z )) ∣ 2 ≤ y − r z E ∣ L X ( 1/2 + r z /2 ) ∣ 2 ≤ C y − r z r z − 2 . The last inequality follows from the prime variance and the absolute higher-power sum computed above. Finite sums converge in L 2 L^2 L 2 Δ / ( 1 − z ) 2 \Delta/(1-z)^2 Δ/ ( 1 − z ) 2 □ \square □
The abstract two-sided order theorem
We state precisely the arithmetic hypotheses used by the order theorem. They concern a fixed number field; none asserts uniformity as the field or its discriminant varies. The measure of the full finite family is kept throughout, including in estimates containing indicators.
Definition 2.19 (Abstract Euler-family hypotheses). Let K K K (5) . For every sufficiently large x x x ( F x , μ x ) (\mathcal{F}_x,\mu_x) ( F x , μ x ) (6) . We impose the following conditions, with constants independent of x x x f f f
(1) Each a f a_f a f { 0 , ± 1 } \{0,\pm1\} { 0 , ± 1 } a f ( O K ) = 1 a_f(\mathcal{O}_K)=1 a f ( O K ) = 1 f f f ℜ s > 1 \Re s>1 ℜ s > 1
L f ( s ) = ∑ a a f ( a ) ( N a ) s = ∏ p ( 1 − a f ( p ) q p − s ) − 1 . L_f(s)=\sum_{\mathfrak a}\frac{a_f(\mathfrak a)}{(N\mathfrak a)^s}=\prod_{\mathfrak p}(1-a_f(\mathfrak p)q_{\mathfrak p}^{-s})^{-1}. L f ( s ) = a ∑ ( N a ) s a f ( a ) = p ∏ ( 1 − a f ( p ) q p − s ) − 1 . (2) There are κ o > 0 \kappa_o>0 κ o > 0 λ ≥ 0 \lambda\ge0 λ ≥ 0
∣ E μ x a f ( a ) − E X ( a ) ∣ ≤ C x − κ o ( N a ) λ , E X ( a ) = 1 { a = b 2 } ∏ p ∣ a θ p . \left|\mathbb{E}_{\mu_x}a_f(\mathfrak a)-\mathbb{E}X(\mathfrak a)\right|\le Cx^{-\kappa_o}(N\mathfrak a)^\lambda,\qquad\mathbb{E}X(\mathfrak a)=\mathbf{1}_{\{\mathfrak a=\mathfrak b^2\}}\prod_{\mathfrak p\mid\mathfrak a}\theta_{\mathfrak p}. ∣ E μ x a f ( a ) − E X ( a ) ∣ ≤ C x − κ o ( N a ) λ , E X ( a ) = 1 { a = b 2 } p ∣ a ∏ θ p . The square condition is on the ideal, rather than its norm. The conductors satisfy, for fixed 0 < b ≤ B 0<b\le B 0 < b ≤ B η c > 0 \eta_c>0 η c > 0
3 ≤ Q f ≤ x B , μ x { Q f < x b } ≤ C x − η c . 3\le Q_f\le x^B,\qquad\mu_x\{Q_f<x^b\}\le Cx^{-\eta_c}. 3 ≤ Q f ≤ x B , μ x { Q f < x b } ≤ C x − η c . (3) There are finite ideal Dirichlet polynomials M f ( s ) M_f(s) M f ( s ) C m , C t , κ m > 0 C_m,C_t,\kappa_m>0 C m , C t , κ m > 0 0 ≤ v ≤ 3 / 2 0\le v\le3/2 0 ≤ v ≤ 3/2 t t t
E μ x ∣ L f ( 1 / 2 + v + i t ) M f ( 1 / 2 + v + i t ) − 1 ∣ 2 ≤ C ( log x ) C m ( 1 + ∣ t ∣ ) C t e − κ m v log x . (38) \mathbb{E}_{\mu_x}\left|L_f(1/2+v+it)M_f(1/2+v+it)-1\right|^2\le C(\log x)^{C_m}(1+|t|)^{C_t}e^{-\kappa_m v\log x}. \tag*{(38)} E μ x ∣ L f ( 1/2 + v + i t ) M f ( 1/2 + v + i t ) − 1 ∣ 2 ≤ C ( log x ) C m ( 1 + ∣ t ∣ ) C t e − κ m v l o g x . (38) In particular the mollifier is holomorphic at the zeros whose detection is required.
(4) For every sufficiently large fixed A A A C d ( A ) C_d(A) C d ( A ) C A C_A C A H = log log x H=\log\log x H = log log x δ = A H / log x \delta=AH/\log x δ = A H / log x
sup ∣ s − 1 / 2 ∣ ≤ 8 δ E μ x ∣ L f ′ ( s ) ∣ 2 ≤ C A e C d ( A ) H . (39) \sup_{\lvert s-1/2\rvert\le8\delta}\mathbb{E}_{\mu_x}\left|L_f'(s)\right|^2\le C_Ae^{C_d(A)H}. \tag*{(39)} ∣ s − 1/2 ∣ ≤ 8 δ sup E μ x L f ′ ( s ) 2 ≤ C A e C d ( A ) H . (39) (5) The function L f L_f L f ℜ s ≥ 1 \Re s\ge1 ℜ s ≥ 1 0 < ℜ ρ < 1 0<\Re\rho<1 0 < ℜ ρ < 1 ρ ↦ 1 − ρ ‾ \rho\mapsto1-\overline{\rho} ρ ↦ 1 − ρ Λ f ( s ) = Q f s / 2 γ f ( s ) L f ( s ) \Lambda_f(s)=Q_f^{s/2}\gamma_f(s)L_f(s) Λ f ( s ) = Q f s /2 γ f ( s ) L f ( s )
Λ f ( s ) = w f Λ f ( 1 − s ) , w f ∈ { 1 , − 1 } . \Lambda_f(s)=w_f\Lambda_f(1-s),\qquad w_f\in\{1,-1\}. Λ f ( s ) = w f Λ f ( 1 − s ) , w f ∈ { 1 , − 1 } . The factor γ f \gamma_f γ f ∣ s − 1 / 2 ∣ ≤ 1 / 4 \lvert s-1/2\rvert\le1/4 ∣ s − 1/2 ∣ ≤ 1/4 γ f ′ / γ f = O ( 1 ) \gamma_f'/\gamma_f=O(1) γ f ′ / γ f = O ( 1 ) f f f 1 / 2 ≤ σ ≤ 2 1/2\le\sigma\le2 1/2 ≤ σ ≤ 2 ∣ t ∣ ≤ 1 \lvert t\rvert\le1 ∣ t ∣ ≤ 1 O ( log ( Q f ( ∣ T ∣ + 2 ) ) ) O(\log(Q_f(\lvert T\rvert+2))) O ( log ( Q f (∣ T ∣ + 2 )))
bounds, and left contours have the decay, needed for the following explicit formulas, which are included as hypotheses:
L f ( s ) = P e ℓ , f ( s ) + ∑ ρ n o n t r i v i a l K ℓ ( ρ − s ) + E t r , f ( s ) , L f = − L f ′ / L f , (40) \mathcal{L}_f(s)=P_{e_\ell,f}(s)+\sum_{\rho\ \mathrm{nontrivial}}K_\ell(\rho-s)+E_{\mathrm{tr},f}(s),\qquad\mathcal{L}_f=-L'_f/L_f, \tag*{(40)} L f ( s ) = P e ℓ , f ( s ) + ρ nontrivial ∑ K ℓ ( ρ − s ) + E tr , f ( s ) , L f = − L f ′ / L f , (40) K ℓ ( z ) = e ℓ z − 2 e 2 ℓ z + e 3 ℓ z ℓ 2 z 3 , ∣ E t r , f ( s ) ∣ ≤ C e − ℓ ℜ s ℓ 2 , K_\ell(z)=\frac{e^{\ell z}-2e^{2\ell z}+e^{3\ell z}}{\ell^2z^3},\qquad\left|E_{\mathrm{tr},f}(s)\right|\leq\frac{Ce^{-\ell\Re s}}{\ell^2}, K ℓ ( z ) = ℓ 2 z 3 e ℓ z − 2 e 2 ℓ z + e 3 ℓ z , ∣ E tr , f ( s ) ∣ ≤ ℓ 2 C e − ℓ ℜ s , ∑ ρ n o n t r i v i a l σ − ℜ ρ ∣ s − ρ ∣ 2 = 1 2 log Q f + O ( 1 ) − ℜ L f ( s ) . (41) \sum_{\rho\ \mathrm{nontrivial}}\frac{\sigma-\Re\rho}{\left|s-\rho\right|^2}=\frac{1}{2}\log Q_f+O(1)-\Re\mathcal{L}_f(s). \tag*{(41)} ρ nontrivial ∑ ∣ s − ρ ∣ 2 σ − ℜ ρ = 2 1 log Q f + O ( 1 ) − ℜ L f ( s ) . (41) Here ℓ ≥ 8 \ell\geq8 ℓ ≥ 8 L f ( s ) ≠ 0 L_f(s)\ne0 L f ( s ) = 0 ℜ s ≥ 1 / 2 \Re s\geq1/2 ℜ s ≥ 1/2 ∣ ℑ s ∣ ≤ 1 |\Im s|\leq1 ∣ℑ s ∣ ≤ 1 1 / 2 ≤ σ ≤ 2 1/2\leq\sigma\leq2 1/2 ≤ σ ≤ 2 ∣ t ∣ ≤ 1 |t|\leq1 ∣ t ∣ ≤ 1
P y , f ( s ) = ∑ N a ≤ y 3 Λ K ( a ) w y ( N a ) a f ( a ) ( N a ) s , Λ K ( p k ) = log N p , P_{y,f}(s)=\sum_{Na\leq y^3}\frac{\Lambda_K(a)w_y(Na)a_f(a)}{(Na)^s},\qquad\Lambda_K(\mathfrak{p}^k)=\log N\mathfrak{p}, P y , f ( s ) = N a ≤ y 3 ∑ ( N a ) s Λ K ( a ) w y ( N a ) a f ( a ) , Λ K ( p k ) = log N p , and Λ K \Lambda_K Λ K
Theorem 2.20 (Abstract two-sided order). Under Definition 2.19 , there are positive constants c c c C C C c − c_- c − C + C_+ C + μ x \mu_x μ x C e − c log log x Ce^{-c\sqrt{\log\log x}} C e − c l o g l o g x
c − log log Q f ≤ N L f ′ d i s t ( [ 1 / 2 , 1 ] ) ≤ N L f ′ m u l t ( [ 1 / 2 , 1 ] ) ≤ C + log log Q f . c_-\log\log Q_f\leq N^{\mathrm{dist}}_{L'_f}([1/2,1])\leq N^{\mathrm{mult}}_{L'_f}([1/2,1])\leq C_+\log\log Q_f. c − log log Q f ≤ N L f ′ dist ([ 1/2 , 1 ]) ≤ N L f ′ mult ([ 1/2 , 1 ]) ≤ C + log log Q f . Zeros at 1 / 2 1/2 1/2 μ x \mu_x μ x
Proof. The independent probabilistic and algebraic inputs have been proved above: analytic nontriviality in Lemma 2.1, smooth Gaussian comparison in Lemma 2.4 , joint growth and negative moments in Lemmas 2.6 and 2.7 , the finite-dimensional polynomial test in Lemma 2.13, the coefficient covariance and tail in Lemma 2.18 , and arithmetic norm transfer in Lemma 2.17 . The analytic assembly from the displayed family hypotheses, including the zero detector, retained and endpoint intervals, and the central functional-equation argument, is proved in Section A. That proof gives both bounds with log log x \log\log x log log x C e − c log log x Ce^{-c\sqrt{\log\log x}} C e − c l o g l o g x Q f < x b Q_f<x^b Q f < x b log log Q f = log log x + O b , B ( 1 ) \log\log Q_f=\log\log x+O_{b,B}(1) log log Q f = log log x + O b , B ( 1 ) O ( x − η c ) O(x^{-\eta c}) O ( x − η c )
Remark 2.21. The hypotheses do not claim that the mollified estimate follows from orthogonality, or that the explicit formulas follow merely from a functional equation. Replacing the logarithmic loss in (38) by x ϵ x^\epsilon x ϵ v = A H / log x v=AH/\log x v = A H / log x A A A A A A
Analytic approximation and local zero counting
We first give the analytic argument for real families under GRH. The arithmetic inputs are the finite-word orthogonality, weighted prime sums, and model density established in the preceding section. All expectations, including those with an indicator of a good event, use the original family denominator.
Normalized analytic hypotheses
Write L X = log X L_X=\log X L X = log X H = log L X H=\log L_X H = log L X L D ( s ) L_D(s) L D ( s )
L D ( s ) = exp ( ∑ p , k ≥ 1 a D , p , k k p k s ) , ∣ a D , p , k ∣ ≤ d , ℜ s > 1. (42) L_D(s)=\exp\left(\sum_{p,k\geq1}\frac{a_{D,p,k}}{kp^{ks}}\right),\qquad|a_{D,p,k}|\leq d,\qquad\Re s>1. \tag*{(42)} L D ( s ) = exp p , k ≥ 1 ∑ k p k s a D , p , k , ∣ a D , p , k ∣ ≤ d , ℜ s > 1. (42) The degree bound d d d L D ′ ≢ 0 L_D'\not\equiv0 L D ′ ≡ 0
Λ D ( s ) = Q D s / 2 ∏ j = 1 d Γ R ( s + μ j ) L D ( s ) , Γ R ( s ) = π − s / 2 Γ ( s / 2 ) , \Lambda_D(s)=Q_D^{s/2}\prod_{j=1}^{d}\Gamma_{\mathbb{R}}(s+\mu_j)L_D(s),\qquad\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2), Λ D ( s ) = Q D s /2 j = 1 ∏ d Γ R ( s + μ j ) L D ( s ) , Γ R ( s ) = π − s /2 Γ ( s /2 ) , is entire of order one and satisfies Λ D ( s ) = ϵ D Λ D ( 1 − s ) \Lambda_D(s)=\epsilon_D\Lambda_D(1-s) Λ D ( s ) = ϵ D Λ D ( 1 − s ) ϵ D ∈ { 1 , − 1 } \epsilon_D\in\{1,-1\} ϵ D ∈ { 1 , − 1 } μ j \mu_j μ j c L X ≤ log Q D ≤ C L X cL_X\leq\log Q_D\leq CL_X c L X ≤ log Q D ≤ C L X
∣ L D ( s ) ∣ ≤ C M [ Q D ( 2 + ∣ ℑ s ∣ ) ] C M ( − M ≤ ℜ s ≤ M ) . (43) |L_D(s)|\leq C_M[Q_D(2+|\Im s|)]^{C_M}\qquad(-M\leq\Re s\leq M). \tag*{(43)} ∣ L D ( s ) ∣ ≤ C M [ Q D ( 2 + ∣ℑ s ∣ ) ] C M ( − M ≤ ℜ s ≤ M ) . (43) GRH in this section means that every zero of Λ D \Lambda_D Λ D ℜ s = 1 / 2 \Re s=1/2 ℜ s = 1/2 L D = − L D ′ / L D \mathcal{L}_D=-L_D'/L_D L D = − L D ′ / L D G D ( w ) = e − w L D ( 1 / 2 + e − w ) G_D(w)=e^{-w}\mathcal{L}_D(1/2+e^{-w}) G D ( w ) = e − w L D ( 1/2 + e − w ) ∣ ℑ w ∣ < π / 2 |\Im w|<\pi/2 ∣ℑ w ∣ < π /2
The smoothing weight below is the classical Selberg weight; see [20 Section 3 and Lemma 3.1] for its quadratic Dirichlet formulation and the zero-dependent anchor. We give the contour and anchor estimates needed for the present real families.
Lemma 3.1 (Paired zero formula and Selberg approximation). Under these hypotheses, if ℓ ≥ 8 \ell\geq8 ℓ ≥ 8 a ≥ 4 / ℓ a\geq4/\ell a ≥ 4/ ℓ a ≤ 3 / 2 a\leq3/2 a ≤ 3/2 ∣ t ∣ ≤ 1 |t|\leq1 ∣ t ∣ ≤ 1
∣ L D ( 1 / 2 + a + i t ) − P ℓ , D ( 1 / 2 + a + i t ) ∣ ≤ C e − a ℓ { L X + ∣ P ℓ , D ( 1 / 2 + 4 / ℓ + i t ) ∣ } . (44) \left|\mathcal{L}_D(1/2+a+it)-P_{\ell,D}(1/2+a+it)\right|\leq Ce^{-a\ell}\left\{L_X+\left|P_{\ell,D}(1/2+4/\ell+it)\right|\right\}. \tag*{(44)} ∣ L D ( 1/2 + a + i t ) − P ℓ , D ( 1/2 + a + i t ) ∣ ≤ C e − a ℓ { L X + ∣ P ℓ , D ( 1/2 + 4/ ℓ + i t ) ∣ } . (44) Here
P ℓ , D ( s ) = ∑ p k ≤ e 3 ℓ w ℓ ( p k ) a D , p , k ( log p ) p − k s , P_{\ell,D}(s)=\sum_{p^k\leq e^{3\ell}}w_{\ell}(p^k)a_{D,p,k}(\log p)p^{-ks}, P ℓ , D ( s ) = p k ≤ e 3 ℓ ∑ w ℓ ( p k ) a D , p , k ( log p ) p − k s , and, with x = log n / ℓ x=\log n/\ell x = log n / ℓ
w ℓ ( n ) = 1 2 { ( 3 − x ) + 2 − 2 ( 2 − x ) + 2 + ( 1 − x ) + 2 } . (45) w_{\ell}(n)=\frac{1}{2}\{(3-x)_+^2-2(2-x)_+^2+(1-x)_+^2\}. \tag*{(45)} w ℓ ( n ) = 2 1 {( 3 − x ) + 2 − 2 ( 2 − x ) + 2 + ( 1 − x ) + 2 } . (45) In particular 0 ≤ w ℓ ≤ 1 0\leq w_{\ell}\leq1 0 ≤ w ℓ ≤ 1 w ℓ = 1 w_{\ell}=1 w ℓ = 1 n ≤ e ℓ n\leq e^\ell n ≤ e ℓ
Proof. Normalization gives ∣ L D ( 2 + i T ) ∣ ≥ exp ( − d ∑ p , k p − 2 k / k ) > 0 |L_D(2+iT)|\geq\exp(-d\sum_{p,k}p^{-2k}/k)>0 ∣ L D ( 2 + i T ) ∣ ≥ exp ( − d ∑ p , k p − 2 k / k ) > 0 2 + i T 2+iT 2 + i T 2 2 2 3 3 3 (43) , bounds the number of critical-line zeros with ∣ γ − T ∣ ≤ 1 |\gamma-T|\leq1 ∣ γ − T ∣ ≤ 1 C log ( Q D ( ∣ T ∣ + 2 ) ) C\log(Q_D(|T|+2)) C log ( Q D ( ∣ T ∣ + 2 )) L D L_D L D
Write z = s − 1 / 2 z=s-1/2 z = s − 1/2 α , − α \alpha,-\alpha α , − α P D \mathcal{P}_D P D m D m_D m D e a z + b e^{az+b} e a z + b a = 0 a=0 a = 0
L D ( 1 / 2 + z ) = B D ( z ) − m D z − ∑ α ∈ P D 2 z z 2 − α 2 , B D ( z ) = 1 2 log Q D + ∑ j Γ R ′ Γ R ( 1 / 2 + z + μ j ) . (46) \mathcal{L}_D(1/2+z)=B_D(z)-\frac{m_D}{z}-\sum_{\alpha\in\mathcal{P}_D}\frac{2z}{z^2-\alpha^2},\qquad B_D(z)=\frac{1}{2}\log Q_D+\sum_j\frac{\Gamma_{\mathbb{R}}'}{\Gamma_{\mathbb{R}}}(1/2+z+\mu_j). \tag*{(46)} L D ( 1/2 + z ) = B D ( z ) − z m D − α ∈ P D ∑ z 2 − α 2 2 z , B D ( z ) = 2 1 log Q D + j ∑ Γ R Γ R ′ ( 1/2 + z + μ j ) . (46) The function B D B_D B D O ( L X ) O(L_X) O ( L X ) ∣ z ∣ ≤ 1 / 4 |z|\leq1/4 ∣ z ∣ ≤ 1/4 0 < a ≤ 3 / 2 0<a\leq3/2 0 < a ≤ 3/2 ∣ t ∣ ≤ 1 |t|\leq1 ∣ t ∣ ≤ 1 1 2 log Q D + O ( 1 ) \frac{1}{2}\log Q_D+O(1) 2 1 log Q D + O ( 1 ) (46) becomes the positive identity
∑ ρ a a 2 + ( t − γ ) 2 = ℜ B D ( a + i t ) − ℜ L D ( 1 / 2 + a + i t ) . (47) \sum_{\rho}\frac{a}{a^2+(t-\gamma)^2}=\Re B_D(a+it)-\Re\mathcal{L}_D(1/2+a+it). \tag*{(47)} ρ ∑ a 2 + ( t − γ ) 2 a = ℜ B D ( a + i t ) − ℜ L D ( 1/2 + a + i t ) . (47) Zeros in this sum are counted with multiplicity, including γ = 0 \gamma=0 γ = 0
For completeness the smoothing kernel is
K ℓ ( z ) = e ℓ z − 2 e 2 ℓ z + e 3 ℓ z ℓ 2 z 3 . K_{\ell}(z)=\frac{e^{\ell z}-2e^{2\ell z}+e^{3\ell z}}{\ell^{2}z^{3}}. K ℓ ( z ) = ℓ 2 z 3 e ℓ z − 2 e 2 ℓ z + e 3 ℓ z . Inverse Mellin inversion of ( u + ) 2 / 2 (u_{+})^{2}/2 ( u + ) 2 /2 (45) ; the identity ( 3 − x ) 2 − 2 ( 2 − x ) 2 + ( 1 − x ) 2 = 2 (3-x)^{2}-2(2-x)^{2}+(1-x)^{2}=2 ( 3 − x ) 2 − 2 ( 2 − x ) 2 + ( 1 − x ) 2 = 2 n n n
L D ( s ) = P ℓ , D ( s ) + ∑ ρ K ℓ ( ρ − s ) + E t r , D ( s ) , ∣ E t r , D ( s ) ∣ ≤ C ℓ − 2 e − ℓ ℜ s . (48) \mathcal{L}_{D}(s)=P_{\ell,D}(s)+\sum_{\rho}K_{\ell}(\rho-s)+E_{\mathrm{tr},D}(s),\qquad|E_{\mathrm{tr},D}(s)|\leq C\ell^{-2}e^{-\ell\Re s}. \tag*{(48)} L D ( s ) = P ℓ , D ( s ) + ρ ∑ K ℓ ( ρ − s ) + E tr , D ( s ) , ∣ E tr , D ( s ) ∣ ≤ C ℓ − 2 e − ℓ ℜ s . (48) Here the last sum of residues is from the fixed gamma progressions − μ j − 2 n -\mu_{j}-2n − μ j − 2 n K ℓ ( ρ − s ) K_{\ell}(\rho-s) K ℓ ( ρ − s ) + 1 +1 + 1
We justify the contour limits explicitly. In each unit interval at large height choose a horizontal line at distance at least c / log ( Q D ( T + 2 ) ) c/\log(Q_{D}(T+2)) c / log ( Q D ( T + 2 )) 2 + i t 2+it 2 + i t O M ( log 2 ( Q D ( T + 2 ) ) ) O_{M}(\log^{2}(Q_{D}(T+2))) O M ( log 2 ( Q D ( T + 2 ))) ∣ T ∣ − 3 |T|^{-3} ∣ T ∣ − 3 O ( log ( Q D ( M + ∣ t ∣ + 2 ) ) ) O(\log(Q_{D}(M+|t|+2))) O ( log ( Q D ( M + ∣ t ∣ + 2 )))
Set a 0 = 4 / ℓ a_{0}=4/\ell a 0 = 4/ ℓ u u u
∣ K ℓ ( − a 0 + i u ) ∣ ≤ 4 e − 4 ℓ 2 ( a 0 2 + u 2 ) 3 / 2 ≤ e − 4 4 a 0 a 0 2 + u 2 . |K_{\ell}(-a_{0}+iu)|\leq\frac{4e^{-4}}{\ell^{2}(a_{0}^{2}+u^{2})^{3/2}}\leq\frac{e^{-4}}{4}\frac{a_{0}}{a_{0}^{2}+u^{2}}. ∣ K ℓ ( − a 0 + i u ) ∣ ≤ ℓ 2 ( a 0 2 + u 2 ) 3/2 4 e − 4 ≤ 4 e − 4 a 0 2 + u 2 a 0 . Use (47) at a 0 a_{0} a 0 (48) . Since e − 4 / 4 < 1 e^{-4}/4<1 e − 4 /4 < 1 ∣ L D ( 1 / 2 + a 0 + i t ) ∣ ≤ C ( L X + ∣ P ℓ , D ( 1 / 2 + a 0 + i t ) ∣ ) |\mathcal{L}_{D}(1/2+a_{0}+it)|\leq C(L_{X}+|P_{\ell,D}(1/2+a_{0}+it)|) ∣ L D ( 1/2 + a 0 + i t ) ∣ ≤ C ( L X + ∣ P ℓ , D ( 1/2 + a 0 + i t ) ∣ ) a ≥ a 0 a\geq a_{0} a ≥ a 0
∣ K ℓ ( − a + i u ) ∣ ≤ 4 e − a ℓ ℓ 2 a a 0 a 0 a 0 2 + u 2 . |K_{\ell}(-a+iu)|\leq\frac{4e^{-a\ell}}{\ell^{2}aa_{0}}\frac{a_{0}}{a_{0}^{2}+u^{2}}. ∣ K ℓ ( − a + i u ) ∣ ≤ ℓ 2 a a 0 4 e − a ℓ a 0 2 + u 2 a 0 . Another use of the positive identity, and ℓ 2 a a 0 ≥ 16 \ell^{2}aa_{0}\geq16 ℓ 2 a a 0 ≥ 16 (44) . □
Normalized polynomial moments
We will repeatedly use the following consequence of finite-word transfer. In a fixed sector ∣ arg a ∣ ≤ α < π / 2 |\arg a|\leq\alpha<\pi/2 ∣ arg a ∣ ≤ α < π /2 a P ℓ , D ( 1 / 2 + a ) aP_{\ell,D}(1/2+a) a P ℓ , D ( 1/2 + a ) ℓ = o ( L X ) \ell=o(L_{X}) ℓ = o ( L X ) O q ( X − κ e C q ℓ ) O_{q}(X^{-\kappa}e^{C_{q}\ell}) O q ( X − κ e C q ℓ )
At the auxiliary anchor a 0 = 4 / ℓ a_{0}=4/\ell a 0 = 4/ ℓ t t t ℓ − 1 P ℓ , E ( 1 / 2 + a 0 + i t ) \ell^{-1}P_{\ell,E}(1/2+a_{0}+it) ℓ − 1 P ℓ , E ( 1/2 + a 0 + i t )
C ℓ − 1 ( log p ) p − 1 / 2 − a 0 / ( 1 − p − 1 / 2 − a 0 ) . C\ell^{-1}(\log p)p^{-1/2-a_{0}}/(1-p^{-1/2-a_{0}}). C ℓ − 1 ( log p ) p − 1/2 − a 0 / ( 1 − p − 1/2 − a 0 ) . The sum of its squares is O ( ℓ − 2 a 0 − 2 ) = O ( 1 ) O(\ell^{-2}a_{0}^{-2})=O(1) O ( ℓ − 2 a 0 − 2 ) = O ( 1 ) k ≥ 2 k\geq2 k ≥ 2 O ( ℓ − 1 a 0 − 1 ) = O ( 1 ) O(\ell^{-1}a_{0}^{-1})=O(1) O ( ℓ − 1 a 0 − 1 ) = O ( 1 ) O ( ℓ − 1 ) O(\ell^{-1}) O ( ℓ − 1 ) ∥ P ℓ , E ( 1 / 2 + a 0 + i t ) ∥ q ≤ C q ℓ \|P_{\ell,E}(1/2+a_{0}+it)\|_{q}\leq C\sqrt{q}\ell ∥ P ℓ , E ( 1/2 + a 0 + i t ) ∥ q ≤ C q ℓ t t t C q C_{q} C q
In particular, after multiplication by ∣ a ∣ |a| ∣ a ∣ (44) gives the precise normalized error
∥ ∣ a ∣ ( L D − P ℓ , D ) ( 1 / 2 + a ) ∥ q ≤ C q e − ℜ ( a ) ℓ ( ∣ a ∣ L X + ∣ a ∣ ℓ ) . (49) \|\,|a|(\mathcal{L}_{D}-P_{\ell,D})(1/2+a)\,\|_{q}\leq C_{q}e^{-\Re(a)\ell}(|a|L_{X}+|a|\ell). \tag*{(49)} ∥ ∣ a ∣ ( L D − P ℓ , D ) ( 1/2 + a ) ∥ q ≤ C q e − ℜ ( a ) ℓ ( ∣ a ∣ L X + ∣ a ∣ ℓ ) . (49) If ℓ = C y L X / H 120 \ell= C_yL_X/H^{120} ℓ = C y L X / H 120 R e ( a ) ℓ ≥ c C y H \mathop{\mathrm{Re}}(a)\ell\ge cC_yH Re ( a ) ℓ ≥ c C y H x = R e ( a ) ℓ x = \mathop{\mathrm{Re}}(a)\ell x = Re ( a ) ℓ C q H 120 x e − x ≤ C q H 121 e − c ′ C y H C_qH^{120}xe^{-x} \le C_qH^{121}e^{-c'C_yH} C q H 120 x e − x ≤ C q H 121 e − c ′ C y H R e ( a ) ℓ → ∞ \mathop{\mathrm{Re}}(a)\ell\to\infty Re ( a ) ℓ → ∞
A local meromorphic Jensen estimate
The next lemma is also useful without GRH and under finite Euler modifications. It is formulated locally so that no unproved global symmetry of a modified function is required.
Lemma 3.2 (Moving-anchor local Jensen). Let F D F_D F D F D ′ ≠ 0 F_D' \ne0 F D ′ = 0 ∣ z ∣ ≤ 1 / 4 |z| \le1/4 ∣ z ∣ ≤ 1/4
− F D ′ F D ( 1 / 2 + z ) = B D ∗ ( z ) − m D z − ∑ α ∈ P D 2 z z 2 − α 2 , -\frac{F_D'}{F_D(1/2+z)}=B_D^*(z)-\frac{m_D}{z}-\sum_{\alpha\in\mathcal{P}_D}\frac{2z}{z^2-\alpha^2}, − F D ( 1/2 + z ) F D ′ = B D ∗ ( z ) − z m D − α ∈ P D ∑ z 2 − α 2 2 z , where B D ∗ B_D^* B D ∗ O ( L X ) O(L_X) O ( L X ) F D F_D F D Δ = ν / L X \Delta=\nu/L_X Δ = ν / L X 1 ≪ ν = o ( H ) 1\ll\nu=o(H) 1 ≪ ν = o ( H )
N D = m D + 2 # { α : ∣ α ∣ ≤ 12 Δ } , W D = Δ 2 ∑ ∣ α ∣ > 12 Δ ∣ α ∣ − 2 . N_D=m_D+2\#\{\alpha:|\alpha|\le12\Delta\},\qquad W_D=\Delta^2\sum_{|\alpha|>12\Delta}|\alpha|^{-2}. N D = m D + 2# { α : ∣ α ∣ ≤ 12Δ } , W D = Δ 2 ∣ α ∣ > 12Δ ∑ ∣ α ∣ − 2 . If E ( N D + W D ) ≤ C ( ν + 1 ) \mathbb{E}(N_D+W_D)\le C(\nu+1) E ( N D + W D ) ≤ C ( ν + 1 ) U U U [ 1 , 2 ] [1,2] [ 1 , 2 ] A D , U = − Δ F D ′ / F D ( 1 / 2 + U Δ ) A_{D,U}=-\Delta F_D'/F_D(1/2+U\Delta) A D , U = − Δ F D ′ / F D ( 1/2 + U Δ ) a , η > 0 a,\eta>0 a , η > 0
P { R F D ′ m u l t ( [ 1 / 2 , 1 / 2 + Δ ] ) > η H } ≤ P { ∣ A D , U ∣ < e − a } + C ( ν + 1 + log ( ν + 2 ) + a ) η H . (50) \mathbb{P}\{R_{F_D'}^{\mathrm{mult}}([1/2,1/2+\Delta])>\eta H\}\le\mathbb{P}\{|A_{D,U}|<e^{-a}\}+\frac{C(\nu+1+\log(\nu+2)+a)}{\eta H}. \tag*{(50)} P { R F D ′ mult ([ 1/2 , 1/2 + Δ ]) > η H } ≤ P { ∣ A D , U ∣ < e − a } + η H C ( ν + 1 + log ( ν + 2 ) + a ) . (50) Restricted estimates on a good set hold with its indicator and the original denominator.
Proof. Take c = U Δ c=U\Delta c = U Δ z = c + 4 Δ e i θ z=c+4\Delta e^{i\theta} z = c + 4Δ e i θ 0 < p < 1 0<p<1 0 < p < 1
∫ 0 2 π ∣ Δ z − α ∣ p d θ 2 π ≤ C p \int_0^{2\pi}\left|\frac{\Delta}{z-\alpha}\right|^p\frac{d\theta}{2\pi}\le C_p ∫ 0 2 π z − α Δ p 2 π d θ ≤ C p uniformly in α \alpha α p p p C p N D C_pN_D C p N D ∣ z ∣ ≤ 6 Δ |z|\le6\Delta ∣ z ∣ ≤ 6Δ ∣ α ∣ > 12 Δ |\alpha|>12\Delta ∣ α ∣ > 12Δ ∣ z 2 − α 2 ∣ ≥ 3 ∣ α ∣ 2 / 4 |z^2-\alpha^2|\ge3|\alpha|^2/4 ∣ z 2 − α 2 ∣ ≥ 3∣ α ∣ 2 /4 16 W D 16W_D 16 W D O ( ν ) O(\nu) O ( ν ) C log ( 2 + ν + N D + W D ) C\log(2+\nu+N_D+W_D) C log ( 2 + ν + N D + W D ) O ( log ( ν + 2 ) ) O(\log(\nu+2)) O ( log ( ν + 2 ))
Every actual pole in the outer disc is a near zero of F D F_D F D
∫ 1 2 log + 4 ∣ u − π / Δ ∣ d u ≤ 8. \int_1^2\log^+\frac{4}{|u-\pi/\Delta|}\,du\le8. ∫ 1 2 log + ∣ u − π /Δ∣ 4 d u ≤ 8. For example integrate the positive logarithm on the larger interval ∣ u − R e ( π / Δ ) ∣ < 4 |u-\mathop{\mathrm{Re}}(\pi/\Delta)|<4 ∣ u − Re ( π /Δ ) ∣ < 4 8 E N D 8\mathbb{E}N_D 8 E N D 2 Δ 2\Delta 2Δ 4 Δ 4\Delta 4Δ log 2 \log2 log 2 F D F_D F D r r r F D ′ F_D' F D ′ r − 1 r-1 r − 1 F D ′ F_D' F D ′ − F D ′ / F D -F_D'/F_D − F D ′ / F D
R F D ′ m u l t ≤ B D , U + + log − ∣ A D , U ∣ + J D , U log 2 + N D . R_{F_D'}^{\mathrm{mult}}\le\frac{B_{D,U}^{+}+\log^-|A_{D,U}|+J_{D,U}}{\log2}+N_D. R F D ′ mult ≤ log 2 B D , U + + log − ∣ A D , U ∣ + J D , U + N D . The target interval is contained in the inner disc for all U ∈ [ 1 , 2 ] U\in[1,2] U ∈ [ 1 , 2 ] ∣ A D , U ∣ < e − a |A_{D,U}|<e^{-a} ∣ A D , U ∣ < e − a (50) . Boundary poles and zeros follow by radial limits of the circle-average formula; their logarithmic singularities are integrable. Tonelli applies throughout to nonnegative majorants. □
Figure 3. The shifted Jensen construction in the w w w [ t j − 1 / 4 , t j + 1 / 4 ] [t_j-1/4,t_j+1/4] [ t j − 1/4 , t j + 1/4 ] 1 / 2 1/2 1/2 z ∗ ∈ V j z_* \in V_j z ∗ ∈ V j
Eight fixed anchors
In the GRH setting the near poles satisfy a useful deterministic gap: if c ∈ [ Δ , 2 Δ ] c \in[\Delta,2\Delta] c ∈ [ Δ , 2Δ ] ∣ c − i γ ∣ ≥ Δ |c-i\gamma| \ge\Delta ∣ c − iγ ∣ ≥ Δ
Lemma 3.3 (Finite anchors for the central interval and bridge). Fix 0 < ε < 1 / 16 0<\varepsilon<1/16 0 < ε < 1/16
ν = H 1 / 2 + ε / 4 , Δ = ν / L X , ℓ c = L X / H 1 / 2 + ε / 8 , r = H − 1 / 4 , \nu=H^{1/2+\varepsilon/4},\qquad\Delta=\nu/L_X,\qquad\ell_c=L_X/H^{1/2+\varepsilon/8},\qquad r=H^{-1/4}, ν = H 1/2 + ε /4 , Δ = ν / L X , ℓ c = L X / H 1/2 + ε /8 , r = H − 1/4 , T F = H 1 / 4 + ε / 32 , M = ⌈ H 1 / 2 + 3 ε / 32 ⌉ . T_F=H^{1/4+\varepsilon/32},\qquad M=\left\lceil H^{1/2+3\varepsilon/32}\right\rceil. T F = H 1/4 + ε /32 , M = ⌈ H 1/2 + 3 ε /32 ⌉ . For H − 1 / 2 + ε ≤ z ≤ 1 H^{-1/2+\varepsilon}\le z\le1 H − 1/2 + ε ≤ z ≤ 1 L D ′ L'_D L D ′ [ 1 / 2 , 1 / 2 + e 5 H 121 / L X ] [1/2,1/2+e^{5H^{121}/L_X}] [ 1/2 , 1/2 + e 5 H 121 / L X ] C ε z H C_\varepsilon zH C ε z H
C ε H − 2 log H + C ε H − 6 . C_\varepsilon H^{-2}\log H+C_\varepsilon H^{-6}. C ε H − 2 log H + C ε H − 6 . The same conclusion holds with a fixed finite Euler modification of the type treated below.
Proof. First take eight distinct fixed shifts v 1 , … , v 8 v_1,\ldots,v_8 v 1 , … , v 8 δ j = e − t − v j \delta_j=e^{-t-v_j} δ j = e − t − v j [ c Δ , C H 121 / L X ] [c\Delta,CH^{121}/L_X] [ c Δ , C H 121 / L X ] f r ( x ) = exp ( 8 C 0 2 ) exp ( − ∥ x ∥ 2 / r 2 ) f_r(x)=\exp(8C_0^2)\exp(-\|x\|^2/r^2) f r ( x ) = exp ( 8 C 0 2 ) exp ( − ∥ x ∥ 2 / r 2 ) ∣ x j ∣ ≤ C 0 r |x_j|\le C_0r ∣ x j ∣ ≤ C 0 r f ^ ( τ ) = ∫ e − i τ ⋅ x f ( x ) d x \widehat f(\tau)=\int e^{-i\tau\cdot x}f(x)\,dx f ( τ ) = ∫ e − i τ ⋅ x f ( x ) d x L 1 L^1 L 1 ( 2 π ) − 8 ∥ f ^ r ∥ 1 = f r ( 0 ) (2\pi)^{-8}\|\widehat f_r\|_1=f_r(0) ( 2 π ) − 8 ∥ f r ∥ 1 = f r ( 0 ) r r r T F T_F T F O ( e − c ( r T F ) 2 ) O(e^{-c(rT_F)^2}) O ( e − c ( r T F ) 2 )
Δ ℓ c = H ε / 8 , r T F = H ε / 32 , T F / M = O ( H − ε / 64 ) , M ℓ c / L X = O ( H − ε / 32 ) . (51) \Delta\ell_c=H^{\varepsilon/8},\qquad rT_F=H^{\varepsilon/32},\qquad T_F/\sqrt{M}=O(H^{-\varepsilon/64}),\qquad M\ell_c/L_X=O(H^{-\varepsilon/32}). \tag*{(51)} Δ ℓ c = H ε /8 , r T F = H ε /32 , T F / M = O ( H − ε /64 ) , M ℓ c / L X = O ( H − ε /32 ) . (51) Also M log ( 2 + T F ) = o ( L X ) M\log(2+T_F)=o(L_X) M log ( 2 + T F ) = o ( L X ) H − 1 H^{-1} H − 1 H H H exp ( − c H ε / 8 ) \exp(-cH^{\varepsilon/8}) exp ( − c H ε /8 ) O ( r − 1 ) O(r^{-1}) O ( r − 1 )
The smooth Gaussian comparison has maximum distance δ j ≪ H 121 e − H \delta_j\ll H^{121}e^{-H} δ j ≪ H 121 e − H O ( r − j ) O(r^{-j}) O ( r − j ) j ≤ 3 j\le3 j ≤ 3 H − 1 H^{-1} H − 1 ( v / 4 ) sech 2 ( ( v i − v j ) / 2 ) (v/4)\operatorname{sech}^2((v_i-v_j)/2) ( v /4 ) sech 2 (( v i − v j ) /2 ) λ j = e − v j \lambda_j=e^{-v_j} λ j = e − v j λ j e − λ j x \lambda_j e^{-\lambda_j x} λ j e − λ j x L 2 ( x d x ) L^2(x\,dx) L 2 ( x d x )
P { ∣ G D ( t + v j ) ∣ ≤ C r ( 1 ≤ j ≤ 8 ) } ≤ C r 8 + O A ( H − A ) = O ( H − 2 ) . (52) \mathbb{P}\{|G_D(t+v_j)| \le Cr \ (1 \le j \le8)\} \le Cr^8 + O_A(H^{-A}) = O(H^{-2}). \tag*{(52)} P { ∣ G D ( t + v j ) ∣ ≤ C r ( 1 ≤ j ≤ 8 )} ≤ C r 8 + O A ( H − A ) = O ( H − 2 ) . (52) No independence among the eight observations is assumed.
At scale Δ \Delta Δ
K D ( Δ ) = ∑ ρ Δ 2 Δ 2 + γ 2 = Δ { B D ( Δ ) − L D ( 1 / 2 + Δ ) } . K_D(\Delta)=\sum_{\rho}\frac{\Delta^2}{\Delta^2+\gamma^2}=\Delta\{B_D(\Delta)-L_D(1/2+\Delta)\}. K D ( Δ ) = ρ ∑ Δ 2 + γ 2 Δ 2 = Δ { B D ( Δ ) − L D ( 1/2 + Δ )} . The polynomial moments and (49) at cutoff ℓ c \ell_c ℓ c ∥ K D ( Δ ) ∥ q ≤ C q ( ν + 1 ) \lVert K_D(\Delta)\rVert_q\le C_q(\nu+1) ∥ K D ( Δ ) ∥ q ≤ C q ( ν + 1 ) q q q
N D ≤ 145 K D ( Δ ) , W D ≤ ( 145 / 288 ) K D ( Δ ) . N_D\le145K_D(\Delta),\qquad W_D\le(145/288)K_D(\Delta). N D ≤ 145 K D ( Δ ) , W D ≤ ( 145/288 ) K D ( Δ ) . The second constant accounts for the fact that a representative α \alpha α ∥ N D ∥ q + ∥ W D ∥ q ≤ C q ( ν + 1 ) \lVert N_D\rVert_q+\lVert W_D\rVert_q\le C_q(\nu+1) ∥ N D ∥ q + ∥ W D ∥ q ≤ C q ( ν + 1 )
Choose U j = 1 + j / 9 U_j=1+j/9 U j = 1 + j /9 c j = U j Δ c_j=U_j\Delta c j = U j Δ v j = − log U j v_j=-\log U_j v j = − log U j C log ( 2 + ν + N D + W D ) C\log(2+\nu+N_D+W_D) C log ( 2 + ν + N D + W D ) ∣ c j − i γ ∣ ≥ Δ \lvert c_j-i\gamma\rvert\ge\Delta ∣ c j − iγ ∣ ≥ Δ log 8 \log8 log 8 Δ / 2 \Delta/2 Δ/2 ∣ Δ L D ( 1 / 2 + c j ) ∣ ≥ r \lvert\Delta L_D(1/2+c_j)\rvert\ge r ∣ Δ L D ( 1/2 + c j )∣ ≥ r
R D m u l t ( [ 1 / 2 , 1 / 2 + Δ ] ) ≤ C ( 1 + ν + N D + W D + log ( 1 / r ) ) . R_D^{\mathrm{mult}}([1/2,1/2+\Delta])\le C(1+\nu+N_D+W_D+\log(1/r)). R D mult ([ 1/2 , 1/2 + Δ ]) ≤ C ( 1 + ν + N D + W D + log ( 1/ r )) . The probability that no anchor works is O ( H − 2 ) O(H^{-2}) O ( H − 2 ) q q q C q ( ( ν + 1 + log H ) / T ) q C_q((\nu+1+\log H)/T)^q C q (( ν + 1 + log H ) / T ) q C T CT C T
The bridge from Δ \Delta Δ e 5 H 121 / L X e^5H^{121}/L_X e 5 H 121 / L X t t t O ( log H ) O(\log H) O ( log H ) r r r r r r (52) and a union bound cost O ( H − 2 log H ) O(H^{-2}\log H) O ( H − 2 log H ) O ( ( log H ) 2 ) O((\log H)^2) O (( log H ) 2 ) q q q O q ( log H ) O_q(\log H) O q ( log H ) C T CT C T C H − 2 log H + C q ( log H / T ) q CH^{-2}\log H+C_q(\log H/T)^q C H − 2 log H + C q ( log H / T ) q (51) .
Finally choose q = ⌈ 8 / ε ⌉ q=\lceil8/\varepsilon\rceil q = ⌈ 8/ ε ⌉ X → ∞ X\to\infty X → ∞ T = z H T=zH T = z H ν / T ≤ H − 3 ε / 4 \nu/T\le H^{-3\varepsilon/4} ν / T ≤ H − 3 ε /4 T ≫ ( log H ) 2 T\gg(\log H)^2 T ≫ ( log H ) 2 O ε ( H − 6 ) O_\varepsilon(H^{-6}) O ε ( H − 6 )
Retained local moments in the bulk
Let
ℓ b = C y L X / H 120 , d X = H 121 / L X , b = H − 121 log H , t ∗ = 50 log H . (53) \ell_b=C_yL_X/H^{120},\qquad d_X=H^{121}/L_X,\qquad b=H-121\log H,\qquad t_*=50\log H. \tag*{(53)} ℓ b = C y L X / H 120 , d X = H 121 / L X , b = H − 121 log H , t ∗ = 50 log H . (53) Use a fixed-radius disc cover of [ log 2 , b ] [\log2,b] [ log 2 , b ] ∣ ℑ w ∣ < π / 2 \lvert\Im w\rvert<\pi/2 ∣ ℑ w ∣ < π /2 ℜ ( e − w ) ℓ b ≥ c C y H \Re(e^{-w})\ell_b\ge cC_yH ℜ ( e − w ) ℓ b ≥ c C y H X X X
Lemma 3.4 (Local moments on one retained event). There is an event R X R_X R X P ( R X c ) ≪ H − 7 \mathbb{P}(R_X^c)\ll H^{-7} P ( R X c ) ≪ H − 7 1 R X \mathbf{1}_{R_X} 1 R X O ( H ) O(H) O ( H )
Proof. The normalized moments and Cauchy’s formula give the supremum and derivative assertions directly. The uniform density of the full random model, the positive Gaussian test, and finite characteristic-function transfer imply
P { ∣ G D ( t ) ∣ ≤ r } ≤ C r + O A ( H − A ) , H − 8 ≤ r ≤ 1 / 4. (54) \mathbb{P}\{|G_D(t)|\le r\}\le Cr+O_A(H^{-A}),\qquad H^{-8}\le r\le1/4. \tag*{(54)} P { ∣ G D ( t ) ∣ ≤ r } ≤ C r + O A ( H − A ) , H − 8 ≤ r ≤ 1/4. (54) uniformly in t t t H 10 H^{10} H 10 ⌈ H 22 ⌉ \lceil H^{22}\rceil ⌈ H 22 ⌉ H − 1 H^{-1} H − 1 L X L_X L X O ( H − 98 ) O(H^{-98}) O ( H − 98 ) r H 10 ≥ H 2 rH^{10} \ge H^2 r H 10 ≥ H 2 t → ∞ t \to\infty t → ∞
Let m j m_j m j M j M_j M j m j < H − 8 m_j < H^{-8} m j < H − 8 O ( H − 7 ) O(H^{-7}) O ( H − 7 ) V j = log + ( 1 / m j ) ≤ 8 log H V_j = \log^+(1/m_j) \le8\log H V j = log + ( 1/ m j ) ≤ 8 log H
P ( R X ∩ { V j > u } ) ≤ C e − u + O A ( H − A ) ( 0 ≤ u ≤ 8 log H ) . \mathbb{P}(R_X \cap\{V_j > u\}) \le C e^{-u} + O_A(H^{-A}) \qquad(0 \le u \le8\log H). P ( R X ∩ { V j > u }) ≤ C e − u + O A ( H − A ) ( 0 ≤ u ≤ 8 log H ) . Integrating this truncated tail gives E ( 1 R X V j q ) ≤ C q \mathbb{E}(\mathbf{1}_{R_X}V_j^q) \le C_q E ( 1 R X V j q ) ≤ C q q q q C ( M j + V j ) C(M_j + V_j) C ( M j + V j )
Crossing meshes and the uniform quantile estimate
We give the quantitative argument because it identifies both the sampling variance and the loss from the central region.
Two nearby crossings of the Gaussian process
Let g g g r ( t ) = sech 2 ( t / 2 ) r(t) = \operatorname{sech}^2(t/2) r ( t ) = sech 2 ( t /2 ) u u u Lemma 2.3 , its paths may be taken C 2 C^2 C 2 c u = e − u 2 / 2 / ( π 2 ) c_u = e^{-u^2/2}/(\pi\sqrt{2}) c u = e − u 2 /2 / ( π 2 )
Lemma 4.1 (Two-point crossing bound). For intervals I , J I,J I , J
E ( N u ( I ) N u ( J ) ) ≤ C u ∫ I ∫ J min ( ∣ s − t ∣ , 1 ) d s d t . \mathbb{E}(N_u(I)N_u(J)) \le C_u \int_I\int_J \min(|s-t|,1)\,ds\,dt. E ( N u ( I ) N u ( J )) ≤ C u ∫ I ∫ J min ( ∣ s − t ∣ , 1 ) d s d t . For two distinct cells of length h ≤ 1 h \le1 h ≤ 1 C u h 2 C_u h^2 C u h 2 C u h C_u h C u h
Proof. For t > 0 t>0 t > 0 r = r ( t ) r=r(t) r = r ( t ) r ′ 2 = r 2 ( 1 − r ) r'^2=r^2(1-r) r ′2 = r 2 ( 1 − r ) Var g ′ ( 0 ) = 1 / 2 \operatorname{Var}g'(0)=1/2 Var g ′ ( 0 ) = 1/2 g ( 0 ) = g ( t ) = u g(0)=g(t)=u g ( 0 ) = g ( t ) = u − u r ′ / ( 1 + r ) -ur'/(1+r) − u r ′ / ( 1 + r ) u r ′ / ( 1 + r ) ur'/(1+r) u r ′ / ( 1 + r )
v c = 1 2 − r ′ 2 1 − r 2 = ( 1 − r ) ( 1 + 2 r ) 2 ( 1 + r ) . v_c=\frac{1}{2}-\frac{r'^2}{1-r^2}=\frac{(1-r)(1+2r)}{2(1+r)}. v c = 2 1 − 1 − r 2 r ′2 = 2 ( 1 + r ) ( 1 − r ) ( 1 + 2 r ) . Conditional Cauchy–Schwarz bounds the absolute product of the two derivatives in expectation by v c + u 2 r ′ 2 / ( 1 + r ) 2 ≤ C u ( 1 − r ) v_c+u^2r'^2/(1+r)^2\le C_u(1-r) v c + u 2 r ′2 / ( 1 + r ) 2 ≤ C u ( 1 − r ) ( u , u ) (u,u) ( u , u )
p t ( u , u ) = e − u 2 / ( 1 + r ) 2 π 1 − r 2 . p_t(u,u)=\frac{e^{-u^2/(1+r)}}{2\pi\sqrt{1-r^2}}. p t ( u , u ) = 2 π 1 − r 2 e − u 2 / ( 1 + r ) . Thus its product with the conditional derivative moment, denoted ρ 2 ( t ) \rho_2(t) ρ 2 ( t )
ρ 2 ( t ) ≤ C u 1 − r ( t ) ≤ C u min ( t , 1 ) . \rho_2(t)\le C_u\sqrt{1-r(t)}\le C_u\min(t,1). ρ 2 ( t ) ≤ C u 1 − r ( t ) ≤ C u min ( t , 1 ) . To justify the required Rice upper bound without presupposing a second-moment identity, replace the root count by ( 2 δ ) − 1 ∫ I 1 ∣ g − u ∣ < δ ∣ g ′ ∣ (2\delta)^{-1}\int_I\mathbf{1}_{|g-u|<\delta}|g'| ( 2 δ ) − 1 ∫ I 1 ∣ g − u ∣ < δ ∣ g ′ ∣ I , J I,J I , J ρ 2 \rho_2 ρ 2 O ( h 2 ) O(h^2) O ( h 2 ) O ( h ) O(h) O ( h )
The probability p u ( h ) p_u(h) p u ( h ) h h h
p u ( h ) = c u h + O u ( h 3 ) . (55) p_u(h)=c_u h+O_u(h^3). \tag*{(55)} p u ( h ) = c u h + O u ( h 3 ) . (55) Indeed the average ( g ( 0 ) + g ( h ) ) / 2 (g(0)+g(h))/2 ( g ( 0 ) + g ( h )) /2 ( g ( h ) − g ( 0 ) ) / 2 (g(h)-g(0))/2 ( g ( h ) − g ( 0 )) /2 ( 1 + r ( h ) ) / 2 (1+r(h))/2 ( 1 + r ( h )) /2 ( 1 − r ( h ) ) / 2 (1-r(h))/2 ( 1 − r ( h )) /2 u u u u u u O ( E ∣ d i f f e r e n c e ∣ 3 ) = O ( h 3 ) O(\mathbb{E}|\mathrm{difference}|^3)=O(h^3) O ( E ∣ difference ∣ 3 ) = O ( h 3 ) O ( h 2 ) O(h^2) O ( h 2 ) h / ( 2 2 ) + O ( h 3 ) h/(2\sqrt{2})+O(h^3) h / ( 2 2 ) + O ( h 3 ) c u h c_u h c u h
For cells separated by distance R ≥ 1 R\ge1 R ≥ 1 1 − r ( h ) ≍ h 2 1-r(h)\asymp h^2 1 − r ( h ) ≍ h 2 O ( e − R ) O(e^{-R}) O ( e − R ) O ( h − 2 e − R ) O(h^{-2}e^{-R}) O ( h − 2 e − R ) − log det ( I − A A ∗ ) -\log\det(I-AA^*) − log det ( I − A A ∗ ) O ( h − 2 e − R ) O(h^{-2}e^{-R}) O ( h − 2 e − R ) R = 16 log H R=16\log H R = 16 log H h = H − 3 / 2 h=H^{-3/2} h = H − 3/2
Sign tests without a separation restriction
In the bulk [ t ∗ , b ] [t_*,b] [ t ∗ , b ] ρ = H − 5 \rho=H^{-5} ρ = H − 5 [ − K , K ] [-K,K] [ − K , K ] K = H K=H K = H j j j C j ρ − j C_j\rho^{-j} C j ρ − j K K K d ≤ 4 d\le4 d ≤ 4
∥ f ^ ∥ 1 ≤ C K d ρ − 2 d , ∫ ∥ τ ∥ ∞ > T F ∣ f ^ ( τ ) ∣ d τ ≤ C K d ρ − 4 d T F − 2 . (56) \|\widehat f\|_1\le CK^d\rho^{-2d},\qquad\int_{\|\tau\|_\infty>T_F}|\widehat f(\tau)|\,d\tau\le CK^d\rho^{-4d}T_F^{-2}. \tag*{(56)} ∥ f ∥ 1 ≤ C K d ρ − 2 d , ∫ ∥ τ ∥ ∞ > T F ∣ f ( τ ) ∣ d τ ≤ C K d ρ − 4 d T F − 2 . (56) For example apply 1 − ∂ j 2 1-\partial_j^2 1 − ∂ j 2 T F T_F T F O ( K d ) O(K^d) O ( K d ) d = 4 d=4 d = 4 T F = H 50 T_F=H^{50} T F = H 50 M = ⌈ H 102 ⌉ M=\lceil H^{102}\rceil M = ⌈ H 102 ⌉ C H 44 CH^{44} C H 44 C H − 16 CH^{-16} C H − 16
T F / M = O ( H − 1 ) , M ℓ b / L X = O ( H − 18 ) , M log ( 2 + T F ) = o ( L X ) . T_F/\sqrt{M}=O(H^{-1}),\qquad M\ell_b/L_X=O(H^{-18}),\qquad M\log(2+T_F)=o(L_X). T F / M = O ( H − 1 ) , M ℓ b / L X = O ( H − 18 ) , M log ( 2 + T F ) = o ( L X ) . Finite characteristic-function transfer thus pays the entire Fourier norm. Approximation errors are H O ( 1 ) e − c H H^{O(1)}e^{-cH} H O ( 1 ) e − cH C y C_y C y e − t ∗ = H − 50 e^{-t_*}=H^{-50} e − t ∗ = H − 50 H 5 H^5 H 5 H 10 H^{10} H 10 H 15 H^{15} H 15 O ( H − 45 ) O(H^{-45}) O ( H − 45 ) O ( H − 90 ) O(H^{-90}) O ( H − 90 ) O ( H − 135 ) O(H^{-135}) O ( H − 135 ) O ( ρ ) O(\rho) O ( ρ ) O ( K − 8 ) O(K^{-8}) O ( K − 8 ) O ( H − 7 ) O(H^{-7}) O ( H − 7 )
δ H ≤ C H − 5 . (57) \delta_H\le CH^{-5}. \tag*{(57)} δ H ≤ C H − 5 . (57) This proof uses no inverse of the covariance of the observation vector. It remains valid for close or repeated times.
Theorem 4.2 (Quantile estimate for real Euler families under GRH). Under the real model, finite-word, weighted-prime, density, and analytic hypotheses above, for each fixed 0 < ε < 1 / 16 0<\varepsilon<1/16 0 < ε < 1/16 H − 1 / 2 + ε ≤ z ≤ 1 , H^{-1/2+\varepsilon}\le z\le1, H − 1/2 + ε ≤ z ≤ 1 ,
P X { max C ∈ { R d i s t , R m u l t , S } ∣ C − c ( m , v ) H Q ∣ > A ε z H Q } ≤ C ε log H H z 2 . \mathbb{P}_X\left\{\max_{C\in\{R^{\mathrm{dist}},R_{\mathrm{mult}},S\}}|C-c(m,v)H_Q|>A_\varepsilon zH_Q\right\}\le C_\varepsilon\frac{\log H}{Hz^2}. P X { C ∈ { R dist , R mult , S } max ∣ C − c ( m , v ) H Q ∣ > A ε z H Q } ≤ C ε H z 2 log H . The conclusion persists under the finite Euler modifications described below.
Proof. Set h = H − 3 / 2 h=H^{-3/2} h = H − 3/2 R = 16 log H R=16\log H R = 16 log H [ t ∗ , b − 2 ] [t_\ast,b-2] [ t ∗ , b − 2 ]
Let S X m e s h S_X^{\mathrm{mesh}} S X mesh G D G_D G D R X R_X R X O ( H / h ) O(H/h) O ( H / h ) O ( H ) O(H) O ( H ) R R R O ( H ( R + 1 ) ) O(H(R+1)) O ( H ( R + 1 )) H 2 H^2 H 2 O ( h − 4 e − R ) O(h^{-4}e^{-R}) O ( h − 4 e − R ) O ( δ H / h 2 ) O(\delta_H/h^2) O ( δ H / h 2 )
Var S X m e s h H 2 ≪ log H H + h − 4 e − R + δ H h 2 ≪ log H H . (58) \frac{\operatorname{Var} S_X^{\mathrm{mesh}}}{H^2}\ll\frac{\log H}{H}+h^{-4}e^{-R}+\frac{\delta_H}{h^2}\ll\frac{\log H}{H}. \tag*{(58)} H 2 Var S X mesh ≪ H log H + h − 4 e − R + h 2 δ H ≪ H log H . (58) This is a variance bound for a bounded mesh statistic. It does not bound the variance of the full root count. From (55) and (57) ,
∣ E S X m e s h H − c ( m , v ) ∣ ≪ h 2 + log H H + δ H h ≪ log H H . (59) \left|\frac{\mathbb{E}S_X^{\mathrm{mesh}}}{H}-c(m,v)\right|\ll h^2+\frac{\log H}{H}+\frac{\delta_H}{h}\ll\frac{\log H}{H}. \tag*{(59)} H E S X mesh − c ( m , v ) ≪ h 2 + H log H + h δ H ≪ H log H . (59) Here the Gaussian level is u = − m / v u=-m/\sqrt{v} u = − m / v v / 2 \sqrt{v}/2 v /2
It remains to bound roots missed by the mesh, including multiple roots. On a cell containing at least two roots counted with multiplicity, Rolle’s theorem and Taylor’s formula give
∣ G D ( t i ) ∣ ≤ C K 1 h 2 or sup I i ∣ G D ′ ′ ∣ > K 1 , K 1 = h − 1 / 8 . |G_D(t_i)|\le CK_1h^2\quad\text{or}\quad\sup_{I_i}|G_D''|>K_1,\qquad K_1=h^{-1/8}. ∣ G D ( t i ) ∣ ≤ C K 1 h 2 or I i sup ∣ G D ′′ ∣ > K 1 , K 1 = h − 1/8 . The assertion includes a double zero, where the first derivative vanishes. The small-value radius is C h 15 / 8 = C H − 45 / 16 Ch^{15/8}=CH^{-45/16} C h 15/8 = C H − 45/16 H − 8 H^{-8} H − 8 (54) and the sixteenth derivative moment bound the event by C h 15 / 8 + C h 2 ≪ h 15 / 8 Ch^{15/8}+Ch^2\ll h^{15/8} C h 15/8 + C h 2 ≪ h 15/8
E ( 1 R X N ( I i ) 1 N ( I i ) ≥ 2 ) ≪ ( h 15 / 8 ) 15 / 16 = h 225 / 128 . \mathbb{E}\left(1_{R_X}N(I_i)1_{N(I_i)\ge2}\right)\ll\left(h^{15/8}\right)^{15/16}=h^{225/128}. E ( 1 R X N ( I i ) 1 N ( I i ) ≥ 2 ) ≪ ( h 15/8 ) 15/16 = h 225/128 . Summing the cells and the O ( log H ) O(\log H) O ( log H ) E X E_X E X
H − 1 E E X ≪ h 97 / 128 + log H / H = H − 291 / 256 + O ( log H / H ) . (60) H^{-1}\mathbb{E}E_X\ll h^{97/128}+\log H/H=H^{-291/256}+O(\log H/H). \tag*{(60)} H − 1 E E X ≪ h 97/128 + log H / H = H − 291/256 + O ( log H / H ) . (60) At the initial segment we use retained local root moments, not a Gaussian limit. A cell with precisely one simple root contributes a crossing; all other discrepancies are bounded by the indicated excess. Endpoint corrections are bounded because the shifted mesh avoids roots.
Uniformly for z ≥ H − 1 / 2 + ϵ z\ge H^{-1/2+\epsilon} z ≥ H − 1/2 + ϵ (59) is o ( z ) o(z) o ( z ) (58) and (60) , give a bulk failure probability at tolerance C z H CzH C z H
C { z − 2 log H H + z − 1 H − 291 / 256 + z − 1 log H H + H − 7 } ≪ log H H z 2 . C\left\{z^{-2}\frac{\log H}{H}+z^{-1}H^{-291/256}+z^{-1}\frac{\log H}{H}+H^{-7}\right\}\ll\frac{\log H}{Hz^2}. C { z − 2 H log H + z − 1 H − 291/256 + z − 1 H log H + H − 7 } ≪ H z 2 log H . For the absorption, the ratio of the second term to the first is at most H − 35 / 256 / log H H^{-35/256}/\log H H − 35/256 / log H z ≤ 1 z\le1 z ≤ 1 Lemma 3.3 bounds the missing central and bridge multiplicity by C ϵ z H C_\epsilon zH C ϵ z H C ϵ z H C_\epsilon zH C ϵ z H c ( m , v ) H c(m,v)H c ( m , v ) H
The strict-sign count follows from Lemma 4.3 below. The bounded conductor comparison ∣ H Q − H ∣ ≤ C |H_Q-H|\le C ∣ H Q − H ∣ ≤ C z H → ∞ zH\to\infty z H → ∞ H Q H_Q H Q
Lemma 4.3 (Closed counts and strict signs). For a nonzero real analytic function on a neighborhood of [ a , b ] [a,b] [ a , b ] q q q M M M S S S
2 q − M − 2 ≤ S ≤ q . 2q-M-2 \le S \le q. 2 q − M − 2 ≤ S ≤ q . Proof. An interior zero has a strict sign change exactly when its multiplicity is odd. If there are e e e M − q ≥ e M-q \ge e M − q ≥ e S ≥ q − 2 − e ≥ 2 q − M − 2 S \ge q-2-e \ge2q-M-2 S ≥ q − 2 − e ≥ 2 q − M − 2 S ≤ q S \le q S ≤ q
Finite Euler changes and summing dyadic families
Proposition 4.4 (Finite Euler stability). Suppose the hypotheses above hold for L D L_D L D
E ( s ) = ∏ p ∈ S ∏ j = 1 d p ( 1 − α p , j p − s ) , ∣ α p , j ∣ ≤ 1. E(s)=\prod_{p\in S}\prod_{j=1}^{d_p}(1-\alpha_{p,j}p^{-s}), \qquad|\alpha_{p,j}|\le1. E ( s ) = p ∈ S ∏ j = 1 ∏ d p ( 1 − α p , j p − s ) , ∣ α p , j ∣ ≤ 1. For a fixed such E E E ( L D E ) ′ (L_DE)' ( L D E ) ′ Q D Q_D Q D E E E
Proof. All zeros of E E E ℜ s ≤ 0 \Re s\le0 ℜ s ≤ 0 g E = − E ′ / E g_E=-E'/E g E = − E ′ / E
− ( L D E ) ′ L D E = L D + g E , G ~ D ( w ) = G D ( w ) + e − w g E ( 1 / 2 + e − w ) . -\frac{(L_DE)'}{L_DE}=\mathcal{L}_D+g_E,\qquad\widetilde{G}_D(w)=G_D(w)+e^{-w}g_E(1/2+e^{-w}). − L D E ( L D E ) ′ = L D + g E , G D ( w ) = G D ( w ) + e − w g E ( 1/2 + e − w ) . The weighted means m , v m,v m , v S S S g E g_E g E O E ( e − ℓ ℜ s ) O_E(e^{-\ell\Re s}) O E ( e − ℓ ℜ s ) p k > e ℓ p^k>e^\ell p k > e ℓ (44) with modified polynomials.
In the central disc replace B D B_D B D (46) by B D + g E ( 1 / 2 + z ) B_D+g_E(1/2+z) B D + g E ( 1/2 + z ) L D L_D L D L D E L_DE L D E O ( 1 ) O(1) O ( 1 )
Finally L D E L_DE L D E + ∞ +\infty + ∞ L D E ≡ 1 L_DE\equiv1 L D E ≡ 1 E E E L D = 1 / E L_D=1/E L D = 1/ E E = 1 E=1 E = 1 L D ′ ≠ 0 L_D'\ne0 L D ′ = 0
For a family of cardinality ≍ X a \asymp X^a ≍ X a a > 0 a>0 a > 0 X \sqrt X X O ( X a / 2 ) O(X^{a/2}) O ( X a /2 ) H b l o c k = H + O ( 1 ) H_{\rm block}=H+O(1) H block = H + O ( 1 ) O ( X a ) O(X^a) O ( X a ) (3) –(4) . An arbitrary function z ( X ) z(X) z ( X )
Three unconditional families
We count zeros of the derivative of the uncompleted L L L F F F R d i s t ( F ) R^{\mathrm{dist}}(F) R dist ( F ) R m u l t ( F ) R^{\mathrm{mult}}(F) R mult ( F ) [ 1 / 2 , 1 ] [1/2,1] [ 1/2 , 1 ] S − ( F ) S^{-}(F) S − ( F ) ( 1 / 2 , 1 ) (1/2,1) ( 1/2 , 1 )
c ∗ = e − 1 / 2 π 2 . c_*=\frac{e^{-1/2}}{\pi\sqrt{2}}. c ∗ = π 2 e − 1/2 . All the probability measures in this section are the original uniform counting measures. Restricting an expectation to a good event always means inserting its indicator, without renormalizing the measure.
Theorem 5.1. Let F ( X ) \mathcal{F}(X) F ( X ) D D D 1 < ∣ D ∣ ≤ X 1<|D|\le X 1 < ∣ D ∣ ≤ X η > 0 \eta>0 η > 0 o η ( X ) o_\eta(X) o η ( X ) C = R d i s t , R m u l t , S − C=R^{\mathrm{dist}},R^{\mathrm{mult}},S^{-} C = R dist , R mult , S −
∣ C ( L ′ ( ⋅ , χ D ) ) − c ∗ log log ∣ D ∣ ∣ ≤ η log log ∣ D ∣ . \left|C\left(L'(\cdot,\chi_D)\right)-c_*\log\log|D|\right|\le\eta\log\log|D|. ∣ C ( L ′ ( ⋅ , χ D ) ) − c ∗ log log ∣ D ∣ ∣ ≤ η log log ∣ D ∣. Both signs of the discriminant and every possible conductor at 2 2 2
Theorem 5.2. Let K = Q ( i ) K=\mathbb{Q}(i) K = Q ( i )
D X = { d ∈ Z [ i ] : ( d , 2 ) = 1 , ( d ) square-free , X < N ( d ) ≤ 2 X } . \mathcal{D}_X=\{d\in\mathbb{Z}[i]:(d,2)=1,\ (d)\ \text{square-free},\ X<N(d)\le2X\}. D X = { d ∈ Z [ i ] : ( d , 2 ) = 1 , ( d ) square-free , X < N ( d ) ≤ 2 X } . For d ∈ D X d\in\mathcal{D}_X d ∈ D X L d ( s ) = L ( s , χ ( 1 + i ) 5 d ) L_d(s)=L(s,\chi^{5d}_{(1+i)}) L d ( s ) = L ( s , χ ( 1 + i ) 5 d ) 32 N ( d ) 32N(d) 32 N ( d ) Q d = 128 N ( d ) Q_d=128N(d) Q d = 128 N ( d ) η > 0 \eta>0 η > 0 o η ( X ) o_\eta(X) o η ( X )
max C = R d i s t , R m u l t , S − ∣ C ( L d ′ ) − c ∗ log log Q d ∣ ≤ η log log Q d . \max_{C=R^{\mathrm{dist}},R^{\mathrm{mult}},S^{-}}\left|C(L'_d)-c_*\log\log Q_d\right|\le\eta\log\log Q_d. C = R dist , R mult , S − max ∣ C ( L d ′ ) − c ∗ log log Q d ∣ ≤ η log log Q d . The same assertion holds with uniform measure on the distinct primitive characters occurring in this family.
Theorem 5.3. Let k k k H k H_k H k k k k
L ( s , f ) = ∏ p ( 1 − λ f ( p ) p − s + p − 2 s ) − 1 . L(s,f)=\prod_p(1-\lambda_f(p)p^{-s}+p^{-2s})^{-1}. L ( s , f ) = p ∏ ( 1 − λ f ( p ) p − s + p − 2 s ) − 1 . For each fixed η > 0 \eta>0 η > 0 o η ( ∣ H k ∣ ) o_\eta(|H_k|) o η ( ∣ H k ∣ )
max C = R d i s t , R m u l t , S − ∣ C ( L ′ ( ⋅ , f ) ) − c ∗ log log ( k 2 ) ∣ ≤ η log log ( k 2 ) . \max_{C=R^{\mathrm{dist}},R^{\mathrm{mult}},S^{-}}\left|C(L'(\cdot,f))-c_*\log\log(k^2)\right|\le\eta\log\log(k^2). C = R dist , R mult , S − max C ( L ′ ( ⋅ , f )) − c ∗ log log ( k 2 ) ≤ η log log ( k 2 ) . The measure is unweighted, and the assertion includes both root numbers i k = 1 i^k=1 i k = 1 i k = − 1 i^k=-1 i k = − 1
No theorem above assumes the Riemann hypothesis, nonvanishing at the central point, or simplicity of a zero. We give the arithmetic and analytic inputs in some detail, since the passage from a mollified moment to the very short interval next to the centre is sensitive to the family and to the range of the shift.
The arithmetic probability spaces
For the rational quadratic family the independent model has variables X p ∈ { 0 , 1 , − 1 } X_p\in\{0,1,-1\} X p ∈ { 0 , 1 , − 1 }
P ( X p = 1 ) = P ( X p = − 1 ) = p 2 ( p + 1 ) , P ( X p = 0 ) = 1 p + 1 . \mathbb{P}(X_p=1)=\mathbb{P}(X_p=-1)=\frac{p}{2(p+1)},\qquad\mathbb{P}(X_p=0)=\frac{1}{p+1}. P ( X p = 1 ) = P ( X p = − 1 ) = 2 ( p + 1 ) p , P ( X p = 0 ) = p + 1 1 . This includes p = 2 p=2 p = 2 n n n
E F ( X ) χ D ( n ) = 1 n = □ ∏ p ∣ n p p + 1 + O ( X − 1 / 2 n ) . (61) \mathbb{E}_{\mathcal{F}(X)}\chi_D(n)=\mathbf{1}_{n=\square}\prod_{p\mid n}\frac{p}{p+1}+O(X^{-1/2}n). \tag*{(61)} E F ( X ) χ D ( n ) = 1 n = □ p ∣ n ∏ p + 1 p + O ( X − 1/2 n ) . (61) Here and below sufficiently large implied constants also cover bounded parameters. One elementary derivation, which will also fix the normalization needed for the mollifier, is as follows. Write D = ϵ c d D=\epsilon cd D = ϵc d d > 0 d>0 d > 0 ϵ = ± 1 \epsilon=\pm1 ϵ = ± 1 c ∈ { 1 , 4 , 8 } c\in\{1,4,8\} c ∈ { 1 , 4 , 8 } d ≡ r ( m o d 8 ) d\equiv r\pmod{8} d ≡ r ( mod 8 )
c = 1 : ϵ r ≡ 1 ( m o d 4 ) ; c = 4 : ϵ r ≡ 3 ( m o d 4 ) ; c = 8 : r arbitrary odd . c=1:\ \epsilon r\equiv1\pmod{4};\qquad c=4:\ \epsilon r\equiv3\pmod{4};\qquad c=8:\ r\text{ arbitrary odd}. c = 1 : ϵr ≡ 1 ( mod 4 ) ; c = 4 : ϵr ≡ 3 ( mod 4 ) ; c = 8 : r arbitrary odd . Expanding μ 2 ( d ) = ∑ α 2 ∣ d μ ( α ) \mu^2(d)=\sum_{\alpha^2\mid d}\mu(\alpha) μ 2 ( d ) = ∑ α 2 ∣ d μ ( α ) Y / π 2 + O ( Y ) Y/\pi^2+O(\sqrt{Y}) Y / π 2 + O ( Y ) Y Y Y n n n 8 8 8 α ≤ Y \alpha\le\sqrt{Y} α ≤ Y O ( Y n ) O(\sqrt{Y}n) O ( Y n ) n n n ( d , n ) = 1 (d,n)=1 ( d , n ) = 1 p / ( p + 1 ) p/(p+1) p / ( p + 1 ) c = 1 , 4 , 8 c=1,4,8 c = 1 , 4 , 8 4 X / π 2 4X/\pi^2 4 X / π 2 X / π 2 X/\pi^2 X / π 2 X / π 2 X/\pi^2 X / π 2 χ D ( 2 ) = ( − 1 ) ( r 2 − 1 ) / 8 \chi_D(2)=(-1)^{(r^2-1)/8} χ D ( 2 ) = ( − 1 ) ( r 2 − 1 ) /8 (61) , including the factor 2 / 3 2/3 2/3 2 2 2
∣ F ( X ) ∣ = 6 X π 2 + O ( X ) . (62) |\mathcal{F}(X)|=\frac{6X}{\pi^2}+O(\sqrt{X}). \tag*{(62)} ∣ F ( X ) ∣ = π 2 6 X + O ( X ) . (62) For K = Q ( i ) K=\mathbb{Q}(i) K = Q ( i ) X p X_{\mathfrak p} X p p p p N p N\mathfrak p N p X ( 1 + i ) = 0 X_{(1+i)}=0 X ( 1 + i ) = 0
∣ D X ∣ = 2 π X 3 ζ K ( 2 ) + O ( X log ( 2 X ) ) , ∣ E D X χ d ( n ) − E X n ∣ ≪ X − 1 / 3 N n . (63) |D_X|=\frac{2\pi X}{3\zeta_K(2)}+O(\sqrt{X}\log(2X)),\qquad\left|\mathbb{E}_{D_X}\chi_d(n)-\mathbb{E}X_n\right|\ll X^{-1/3}Nn. \tag*{(63)} ∣ D X ∣ = 3 ζ K ( 2 ) 2 π X + O ( X log ( 2 X )) , ∣ E D X χ d ( n ) − E X n ∣ ≪ X − 1/3 N n . (63) For completeness, the number of points of one residue class modulo an ideal q \mathfrak q q ( Y , 2 Y ] (Y,2Y] ( Y , 2 Y ] π Y / N q + O ( Y / N q + 1 ) \pi Y/N\mathfrak q+O(\sqrt{Y/N\mathfrak q}+1) π Y / N q + O ( Y / N q + 1 ) q \mathfrak q q n \mathfrak n n O ( Y N n + N n ) O(\sqrt{Y}N\mathfrak n+N\mathfrak n) O ( Y N n + N n ) N a ≤ 2 X N\mathfrak a\le\sqrt{2X} N a ≤ 2 X O ( X N n log ( 2 X ) ) O(\sqrt{X}N\mathfrak n\log(2X)) O ( X N n log ( 2 X ))
π X 2 ∏ p ∣ n ( 1 − ( N p ) − 1 ) ∏ p 2 ∣ n ( 1 − ( N p ) − 2 ) = 2 π X 3 ζ K ( 2 ) ∏ p ∣ n N p N p + 1 . \frac{\pi X}{2}\prod_{\mathfrak p\mid\mathfrak n}\left(1-(N\mathfrak p)^{-1}\right)\prod_{\mathfrak p^2\mid\mathfrak n}\left(1-(N\mathfrak p)^{-2}\right)=\frac{2\pi X}{3\zeta_K(2)}\prod_{\mathfrak p\mid\mathfrak n}\frac{N\mathfrak p}{N\mathfrak p+1}. 2 π X p ∣ n ∏ ( 1 − ( N p ) − 1 ) p 2 ∣ n ∏ ( 1 − ( N p ) − 2 ) = 3 ζ K ( 2 ) 2 π X p ∣ n ∏ N p + 1 N p . The elementary bound 2 ω ( n ) ≤ N n 2^{\omega(n)}\le Nn 2 ω ( n ) ≤ N n ( 1 + i ) (1+i) ( 1 + i ) (63) proves the second.
Gao’s family definition and functional equation give
Λ d ( s ) = ( 128 N ( d ) ) s / 2 ( 2 π ) − s Γ ( s ) L d ( s ) = Λ d ( 1 − s ) . (64) \Lambda_d(s)=(128N(d))^{s/2}(2\pi)^{-s}\Gamma(s)L_d(s)=\Lambda_d(1-s). \tag*{(64)} Λ d ( s ) = ( 128 N ( d ) ) s /2 ( 2 π ) − s Γ ( s ) L d ( s ) = Λ d ( 1 − s ) . (64) and primitivity of conductor ( ( 1 + i ) 5 d ) ((1+i)^5d) (( 1 + i ) 5 d ) [14 §2.3, (2.1)–(2.4)] . Equal primitive characters must have equal conductor ideals, so their labels differ by a unit. The supplementary quadratic law gives ( − 1 / n ) = 1 (-1/\mathfrak n)=1 ( − 1/ n ) = 1 ( i / n ) = − 1 (i/\mathfrak n)=-1 ( i / n ) = − 1 n ≡ − 1 + 2 i ( m o d 4 ) \mathfrak n\equiv-1+2i\pmod{4} n ≡ − 1 + 2 i ( mod 4 ) [13 (2.2)] . The Chinese remainder theorem permits the additional condition n ≡ 1 ( m o d rad ( d ) ) \mathfrak n\equiv1\pmod{\operatorname{rad}(d)} n ≡ 1 ( mod rad ( d )) d ↦ χ d d\mapsto\chi_d d ↦ χ d { d , − d } \{d,-d\} { d , − d }
The weight family has a different local law. If U j = U j ( x / 2 ) U_j=U_j(x/2) U j = U j ( x /2 )
d μ p ( x ) = 1 + p − 1 ( 1 + p − 1 ) 2 − p − 1 x 2 4 − x 2 2 π d x , − 2 ≤ x ≤ 2. (65) d\mu_p(x)=\frac{1+p^{-1}}{(1+p^{-1})^2-p^{-1}x^2}\frac{\sqrt{4-x^2}}{2\pi}\,dx,\qquad-2\le x\le2. \tag*{(65)} d μ p ( x ) = ( 1 + p − 1 ) 2 − p − 1 x 2 1 + p − 1 2 π 4 − x 2 d x , − 2 ≤ x ≤ 2. (65) The identity ∑ j ≥ 0 p − j U 2 j ( x / 2 ) = 1 + p − 1 ( 1 + p − 1 ) 2 − p − 1 x 2 \sum_{j\ge0}p^{-j}U_{2j}(x/2)=\frac{1+p^{-1}}{(1+p^{-1})^2-p^{-1}x^2} ∑ j ≥ 0 p − j U 2 j ( x /2 ) = ( 1 + p − 1 ) 2 − p − 1 x 2 1 + p − 1 E U j ( X p / 2 ) = p − j / 2 \mathbb{E}U_j(X_p/2)=p^{-j/2} E U j ( X p /2 ) = p − j /2 j j j 2 / 3 2/3 2/3 6 6 6 A 1 ( x ) = x A_1(x)=x A 1 ( x ) = x A j ( x ) = U j ( x / 2 ) − U j − 2 ( x / 2 ) A_j(x)=U_j(x/2)-U_{j-2}(x/2) A j ( x ) = U j ( x /2 ) − U j − 2 ( x /2 ) j ≥ 2 j\ge2 j ≥ 2 ∣ A j ∣ ≤ 2 |A_j|\le2 ∣ A j ∣ ≤ 2 E X p 2 = 1 + p − 1 \mathbb{E}X_p^2=1+p^{-1} E X p 2 = 1 + p − 1
E A 2 j ( X p ) = − ( 1 − p − 1 ) p − j + 1 , E A 2 j − 1 ( X p ) = 0. (66) \mathbb{E}A_{2j}(X_p)=-(1-p^{-1})p^{-j+1},\qquad\mathbb{E}A_{2j-1}(X_p)=0. \tag*{(66)} E A 2 j ( X p ) = − ( 1 − p − 1 ) p − j + 1 , E A 2 j − 1 ( X p ) = 0. (66) Hough’s unweighted coefficient estimate says that, for fixed 0 < ϵ < 2 0 < \epsilon< 2 0 < ϵ < 2 n < k 2 − ϵ n < k^{2-\epsilon} n < k 2 − ϵ
1 ∣ H k ∣ ∑ f ∈ H k λ f ( n ) = 1 n = □ n + O ϵ ( k − γ ( ϵ ) ) , ∣ H k ∣ = ( k − 1 ) / 12 + O ( 1 ) . \frac{1}{|H_k|}\sum_{f\in H_k}\lambda_f(n)=\frac{1_{n=\square}}{\sqrt{n}}+O_{\epsilon}(k^{-\gamma(\epsilon)}),\qquad|H_k|=(k-1)/12+O(1). ∣ H k ∣ 1 f ∈ H k ∑ λ f ( n ) = n 1 n = □ + O ϵ ( k − γ ( ϵ ) ) , ∣ H k ∣ = ( k − 1 ) /12 + O ( 1 ) . See [16 Lemma 2.2] . There is no additional factor of n n n + 1 +1 + 1
To multiply finite polynomials, use the Hecke algebra e m e n = ∑ d ∣ ( m , n ) e m n / d 2 e_m e_n=\sum_{d\mid(m,n)}e_{mn/d^2} e m e n = ∑ d ∣ ( m , n ) e mn / d 2
∥ ∑ a n e n ∥ H = ∑ ∣ a n ∣ τ ( n ) n . \left\|\sum a_n e_n\right\|_H=\sum|a_n|\tau(n)n. ∑ a n e n H = ∑ ∣ a n ∣ τ ( n ) n . It is submultiplicative, since ∑ d ∣ ( m , n ) τ ( m n / d 2 ) = τ ( m ) τ ( n ) \sum_{d\mid(m,n)}\tau(mn/d^2)=\tau(m)\tau(n) ∑ d ∣ ( m , n ) τ ( mn / d 2 ) = τ ( m ) τ ( n ) ϵ = 1 \epsilon=1 ϵ = 1 ∣ λ f ( n ) ∣ , ∣ λ X ( n ) ∣ ≤ τ ( n ) |\lambda_f(n)|,|\lambda_X(n)|\leq\tau(n) ∣ λ f ( n ) ∣ , ∣ λ X ( n ) ∣ ≤ τ ( n ) n ≥ k n\geq k n ≥ k κ > 0 \kappa>0 κ > 0
∣ E H k P − E P ( X ) ∣ ≪ k − κ ∥ P ∥ H (67) |\mathbb{E}_{H_k}P-\mathbb{E}P(X)|\ll k^{-\kappa}\|P\|_H \tag*{(67)} ∣ E H k P − E P ( X ) ∣ ≪ k − κ ∥ P ∥ H (67) for every finite Hecke polynomial, with no support restriction. In the two quadratic families the analogous submultiplicative norm is ∑ ∣ a n ∣ n \sum|a_n|n ∑ ∣ a n ∣ n ∑ ∣ a n ∣ N n \sum|a_n|Nn ∑ ∣ a n ∣ N n 1 / 2 1/2 1/2 1 / 3 1/3 1/3 (61) and (63) .
A central mollified bound for all signed discriminants
The following extension of the quadratic mollifier calculation is needed on the complete family, rather than only on χ − 8 d \chi_{-8d} χ − 8 d
Proposition 5.4. Write L = log X L=\log X L = log X H = log L H=\log L H = log L M = X 1 / 100 M=X^{1/100} M = X 1/100 and let Q ( v ) = 0 Q(v)=0 Q ( v ) = 0 for v ≤ 0 v\leq0 v ≤ 0 Q ( v ) = 3 ( 2 v ) 2 − 2 ( 2 v ) 3 Q(v)=3(2v)^2-2(2v)^3 Q ( v ) = 3 ( 2 v ) 2 − 2 ( 2 v ) 3 for 0 < v < 1 / 2 0<v<1/2 0 < v < 1/2 and Q ( v ) = 1 Q(v)=1 Q ( v ) = 1 for v ≥ 1 / 2 v\geq1/2 v ≥ 1/2 The entire mollifier
M D ( s ) = ( 1 − χ D ( 2 ) 2 − s ) ∑ n ≤ M n o d d μ ( n ) Q ( log ( M / n ) / log M ) χ D ( n ) n s M_D(s)=\left(1-\chi_D(2)2^{-s}\right)\sum_{\substack{n\leq M\\ n\ \mathrm{odd}}}\frac{\mu(n)Q(\log(M/n)/\log M)\chi_D(n)}{n^s} M D ( s ) = ( 1 − χ D ( 2 ) 2 − s ) n ≤ M n odd ∑ n s μ ( n ) Q ( log ( M / n ) / log M ) χ D ( n ) satisfies
E F ( X ) ∣ L ( 1 / 2 + i t , χ D ) M D ( 1 / 2 + i t ) ∣ 2 ≪ H 20 ( ∣ t ∣ ≤ 1 2 L − 5 / 6 ) . (68) \mathbb{E}_{F(X)}\left|L(1/2+it,\chi_D)M_D(1/2+it)\right|^2\ll H^{20}\qquad(|t|\leq\tfrac12L^{-5/6}). \tag*{(68)} E F ( X ) ∣ L ( 1/2 + i t , χ D ) M D ( 1/2 + i t ) ∣ 2 ≪ H 20 ( ∣ t ∣ ≤ 2 1 L − 5/6 ) . (68) Proof. We retain the class notation D = ϵ c d D=\epsilon cd D = ϵc d d ≡ r ( m o d 8 ) d\equiv r\pmod8 d ≡ r ( mod 8 ) Y < d < 2 Y Y<d<2Y Y < d < 2 Y a = ( 1 − ϵ ) / 2 a=(1-\epsilon)/2 a = ( 1 − ϵ ) /2 ζ = χ D ( 2 ) \zeta=\chi_D(2) ζ = χ D ( 2 ) ( c d / π ) ( s + a ) / 2 Γ ( ( s + a ) / 2 ) L ( s , χ D ) (cd/\pi)^{(s+a)/2}\Gamma((s+a)/2)L(s,\chi_D) ( c d / π ) ( s + a ) /2 Γ (( s + a ) /2 ) L ( s , χ D ) + 1 +1 + 1 2 2 2
Here are the modifications of the calculation in [10 §§3–6] . They are recorded also to identify the secondary terms which would be lost by replacing a fixed congruence class by the sum of its classes. For odd n n n τ j ( n ) = ∑ b m o d n ( b / n ) e ( j b / n ) \tau_j(n)=\sum_{b\bmod n}(b/n)e(jb/n) τ j ( n ) = ∑ b mod n ( b / n ) e ( j b / n ) F ^ ( ξ ) = ∫ F ( v ) e ( − v ξ ) d v \widehat{F}(\xi)=\int F(v)e(-v\xi)\,dv F ( ξ ) = ∫ F ( v ) e ( − v ξ ) d v
∑ d ≡ r ( 8 ) ( d n ) F ( d / Y ) = Y 8 n ( 8 n ) ∑ j ∈ Z e ( j r n / 8 ) τ j ( n ) F ^ ( j Y 8 n ) . (69) \sum_{d\equiv r(8)}\left(\frac{d}{n}\right)F(d/Y)=\frac{Y}{8n}\left(\frac{8}{n}\right)\sum_{j\in\mathbb{Z}}e(jrn/8)\tau_j(n)\widehat{F}\left(\frac{jY}{8n}\right). \tag*{(69)} d ≡ r ( 8 ) ∑ ( n d ) F ( d / Y ) = 8 n Y ( n 8 ) j ∈ Z ∑ e ( j r n /8 ) τ j ( n ) F ( 8 n j Y ) . (69) After truncating the square-free expansion at α ≤ V \alpha\leq V α ≤ V
( 8 / n ) ( 8 n ) ∑ α ≤ V ( α , 2 n ) = 1 μ ( α ) α 2 , \frac{(8/n)}{(8n)}\sum_{\substack{\alpha\leq V\\(\alpha,2n)=1}}\frac{\mu(\alpha)}{\alpha^2}, ( 8 n ) ( 8/ n ) α ≤ V ( α , 2 n ) = 1 ∑ α 2 μ ( α ) , with Fourier argument j Y / ( 8 α 2 n ) jY/(8\alpha^2n) j Y / ( 8 α 2 n ) ( ϵ c / n ) (\epsilon c/n) ( ϵc / n ) χ D ( n ) \chi_D(n) χ D ( n ) ( 8 ϵ c / n ) (8\epsilon c/n) ( 8 ϵc / n )
Set A = ( 1 + i ) / 2 A=(1+i)/2 A = ( 1 + i ) /2 B = ( 1 − i ) / 2 B=(1-i)/2 B = ( 1 − i ) /2 ϵ n = 1 \epsilon_n=1 ϵ n = 1 i i i n ≡ 1 n\equiv1 n ≡ 1 3 ( m o d 4 ) 3\pmod{4} 3 ( mod 4 ) G j ( n ) = τ j ( n ) / ϵ n G_j(n)=\tau_j(n)/\epsilon_n G j ( n ) = τ j ( n ) / ϵ n e ( j r n / 8 ) ϵ n e(jrn/8)\epsilon_n e ( j r n /8 ) ϵ n
{ χ 8 ( j r ) e i π / 4 ( 2 s / n ) , j odd , s = χ 4 ( j r ) , i j r / 2 { B + A χ 4 ( n ) } , j ≡ 2 ( m o d 4 ) , ( − 1 ) j / 4 { A + B χ 4 ( n ) } , 4 ∣ j . \begin{cases}
\chi_8(jr)e^{i\pi/4}(2s/n), & j\ \text{odd},\ s=\chi_4(jr),\\
i^{jr/2}\{B+A\chi_4(n)\}, & j\equiv2\pmod{4},\\
(-1)^{j/4}\{A+B\chi_4(n)\}, & 4\mid j.
\end{cases} ⎩ ⎨ ⎧ χ 8 ( j r ) e iπ /4 ( 2 s / n ) , i j r /2 { B + A χ 4 ( n )} , ( − 1 ) j /4 { A + B χ 4 ( n )} , j odd , s = χ 4 ( j r ) , j ≡ 2 ( mod 4 ) , 4 ∣ j . At an unramified odd prime, G j ( p ) = ( j / p ) p G_j(p)=(j/p)\sqrt{p} G j ( p ) = ( j / p ) p
Table 1 (PDF p. 37).
Indeed, the mod-8 phases above can only alter the sign and the power of 2 in the radicand, never a nontrivial odd square-free part.
Let T ϵ ( u ) = cos ( π u / 2 ) + ϵ sin ( π u / 2 ) T_\epsilon(u)=\cos(\pi u/2)+\epsilon\sin(\pi u/2) T ϵ ( u ) = cos ( π u /2 ) + ϵ sin ( π u /2 )
S 8 ( u ) = − 2 − 2 u ( 1 − 2 1 − 2 u ) 1 − 2 − 2 u T ϵ ( u ) , S_8(u)=-\frac{2^{-2u}(1-2^{1-2u})}{1-2^{-2u}}T_\epsilon(u), S 8 ( u ) = − 1 − 2 − 2 u 2 − 2 u ( 1 − 2 1 − 2 u ) T ϵ ( u ) , S 4 ( u ) = 2 u S 8 ( u ) , S 1 ( u ) = ( 2 ζ + 2 − u 1 − 2 − 2 u ) T ϵ ( u ) . (70) S_4(u)=2^uS_8(u),\qquad S_1(u)=\left(\sqrt{2}\,\zeta+\frac{2^{-u}}{1-2^{-2u}}\right)T_\epsilon(u). \tag*{(70)} S 4 ( u ) = 2 u S 8 ( u ) , S 1 ( u ) = ( 2 ζ + 1 − 2 − 2 u 2 − u ) T ϵ ( u ) . (70) For example the c = 8 c=8 c = 8 − A -A − A − B -B − B ϵ \epsilon ϵ − ϵ -\epsilon − ϵ A A A B B B c = 4 c=4 c = 4 c = 1 c=1 c = 1 ζ e i π / 4 \zeta e^{i\pi/4} ζ e iπ /4 ζ e − i π / 4 \zeta e^{-i\pi/4} ζ e − iπ /4 A A A (70) .
Put τ = ( δ 1 + δ 2 ) / 2 \tau=(\delta_1+\delta_2)/2 τ = ( δ 1 + δ 2 ) /2 δ = ( δ 1 − δ 2 ) / 2 \delta=(\delta_1-\delta_2)/2 δ = ( δ 1 − δ 2 ) /2
Γ a , z ( s ) = Γ ( ( 1 / 2 + a + s + z ) / 2 ) Γ ( ( 1 / 2 + a + s − z ) / 2 ) . \Gamma_{a,z}(s)=\Gamma((1/2+a+s+z)/2)\Gamma((1/2+a+s-z)/2). Γ a , z ( s ) = Γ (( 1/2 + a + s + z ) /2 ) Γ (( 1/2 + a + s − z ) /2 ) . The parity change in the secondary integral follows from the exact reflection identity
Γ 0 , z ( s ) T + ( s − z ) Γ 1 , z ( s ) T − ( s − z ) = Γ 0 , s ( z ) Γ 1 , s ( z ) . (71) \frac{\Gamma_{0,z}(s)T_+(s-z)}{\Gamma_{1,z}(s)T_-(s-z)}=\frac{\Gamma_{0,s}(z)}{\Gamma_{1,s}(z)}. \tag*{(71)} Γ 1 , z ( s ) T − ( s − z ) Γ 0 , z ( s ) T + ( s − z ) = Γ 1 , s ( z ) Γ 0 , s ( z ) . (71) To check it, apply reflection to p = ( 1 / 2 + s + z ) / 2 p=(1/2+s+z)/2 p = ( 1/2 + s + z ) /2 q = ( 1 / 2 + s − z ) / 2 q=(1/2+s-z)/2 q = ( 1/2 + s − z ) /2 cot ( π q ) \cot(\pi q) cot ( π q ) T + = 2 sin ( π q ) T_+=\sqrt{2}\sin(\pi q) T + = 2 sin ( π q ) T − = 2 cos ( π q ) T_-=\sqrt{2}\cos(\pi q) T − = 2 cos ( π q )
In the odd-conductor case define
B 2 = ( 1 − ζ 2 − 1 / 2 − τ − δ ) ( 1 − ζ 2 − 1 / 2 − τ + δ ) , E 2 ( s ) = B 2 ( 1 − ζ 2 − 1 / 2 − s + δ ) ( 1 − ζ 2 − 1 / 2 − s − δ ) . B_2=(1-\zeta2^{-1/2-\tau-\delta})(1-\zeta2^{-1/2-\tau+\delta}),\qquad
E_2(s)=\frac{B_2}{(1-\zeta2^{-1/2-s+\delta})(1-\zeta2^{-1/2-s-\delta})}. B 2 = ( 1 − ζ 2 − 1/2 − τ − δ ) ( 1 − ζ 2 − 1/2 − τ + δ ) , E 2 ( s ) = ( 1 − ζ 2 − 1/2 − s + δ ) ( 1 − ζ 2 − 1/2 − s − δ ) B 2 . At either secondary residue z = δ z=\delta z = δ z = − δ z=-\delta z = − δ q = 2 s q=2^s q = 2 s t = 2 z t=2^z t = 2 z
R ζ ( s , z ) = 8 − s S 1 ( s − z ) S 8 ( s − z ) E 2 ( s ) = − 2 ζ t − 2 B 2 q + q − 1 − ζ ( 2 t + ( 2 t ) − 1 ) . (72) R_\zeta(s,z)=8^{-s}\frac{S_1(s-z)}{S_8(s-z)}E_2(s)=\frac{-\sqrt{2}\,\zeta t^{-2}B_2}{q+q^{-1}-\zeta\left(\sqrt{2}t+(\sqrt{2}t)^{-1}\right)}. \tag*{(72)} R ζ ( s , z ) = 8 − s S 8 ( s − z ) S 1 ( s − z ) E 2 ( s ) = q + q − 1 − ζ ( 2 t + ( 2 t ) − 1 ) − 2 ζ t − 2 B 2 . (72) The equality with E 2 ( s ) E_2(s) E 2 ( s ) (72) is asserted only for z = ± δ z=\pm\delta z = ± δ z z z R ζ ( s , z ) R_\zeta(s,z) R ζ ( s , z ) s s s c = 4 c=4 c = 4 2 − z 2^{-z} 2 − z c = 8 c=8 c = 8 (72) cancel against the 2-Euler factor
η 2 , s ( 1 + 2 z ) = ( 1 − ( 2 t 2 q 2 ) − 1 ) ( 1 − ( 2 t 2 ) − 1 ) ( 1 − q 2 / ( 2 t 2 ) ) . \eta_{2,s}(1+2z)=(1-(2t^2q^2)^{-1})(1-(2t^2)^{-1})(1-q^2/(2t^2)). η 2 , s ( 1 + 2 z ) = ( 1 − ( 2 t 2 q 2 ) − 1 ) ( 1 − ( 2 t 2 ) − 1 ) ( 1 − q 2 / ( 2 t 2 )) . Their product is the entire, even expression
ζ 2 t − 4 B 2 ( 1 − ( 2 t 2 ) − 1 ) { q + q − 1 + ζ ( ( 2 t ) + ( 2 t ) − 1 ) } . (73) \frac{\zeta}{\sqrt{2}}t^{-4}B_2(1-(2t^2)^{-1})\{q+q^{-1}+\zeta((\sqrt{2}t)+(\sqrt{2}t)^{-1})\}. \tag*{(73)} 2 ζ t − 4 B 2 ( 1 − ( 2 t 2 ) − 1 ) { q + q − 1 + ζ (( 2 t ) + ( 2 t ) − 1 )} . (73) Thus this cancellation removes all the periodic denominator zeros, not just their real representatives.
An additional precaution is necessary in the even-character classes. Removing the square-free condition can produce the principal character when d = l 2 d=l^2 d = l 2
H δ , τ ( s ) = { s 2 − ( 1 / 2 + δ ) 2 } { s 2 − ( 1 / 2 − δ ) 2 } { τ 2 − ( 1 / 2 + δ ) 2 } { τ 2 − ( 1 / 2 − δ ) 2 } (74) H_{\delta,\tau}(s)=\frac{\{s^2-(1/2+\delta)^2\}\{s^2-(1/2-\delta)^2\}}{\{\tau^2-(1/2+\delta)^2\}\{\tau^2-(1/2-\delta)^2\}} \tag*{(74)} H δ , τ ( s ) = { τ 2 − ( 1/2 + δ ) 2 } { τ 2 − ( 1/2 − δ ) 2 } { s 2 − ( 1/2 + δ ) 2 } { s 2 − ( 1/2 − δ ) 2 } (74) in the approximate functional equation, whose Mellin kernel is Γ a , δ ( s ) H δ , τ ( s ) 2 s / ( s 2 − τ 2 ) \Gamma_{a,\delta}(s)H_{\delta,\tau}(s)2^s/(s^2-\tau^2) Γ a , δ ( s ) H δ , τ ( s ) 2 s / ( s 2 − τ 2 ) H ( τ ) = H ( − τ ) = 1 H(\tau)=H(-\tau)=1 H ( τ ) = H ( − τ ) = 1 E c ( s ) E_c(s) E c ( s ) c = 1 c=1 c = 1 B 2 B_2 B 2 L D ( 2 ) L_D^{(2)} L D ( 2 ) ± ( 1 / 2 ± δ ) \pm(1/2\pm\delta) ± ( 1/2 ± δ ) Γ 0 , s ( z ) \Gamma_{0,s}(z) Γ 0 , s ( z ) s = ± ( 1 / 2 + z ) s=\pm(1/2+z) s = ± ( 1/2 + z ) (73) when ζ = − 1 \zeta=-1 ζ = − 1 H H H ζ = 1 \zeta=1 ζ = 1 a = 1 a=1 a = 1 ∣ ℜ s ∣ < 3 / 4 |\Re s|<3/4 ∣ℜ s ∣ < 3/4
Here is a precise form of the resulting four main terms. Define r v ( n ) = ∑ a b = n ( a / b ) v r_v(n)=\sum_{ab=n}(a/b)^v r v ( n ) = ∑ ab = n ( a / b ) v Z ( s ; v ) = ζ ( s − 2 v ) ζ ( s ) ζ ( s + 2 v ) Z(s;v)=\zeta(s-2v)\zeta(s)\zeta(s+2v) Z ( s ; v ) = ζ ( s − 2 v ) ζ ( s ) ζ ( s + 2 v ) l = l 1 l 2 2 l=l_1l_2^2 l = l 1 l 2 2 l 1 l_1 l 1 Ψ ~ ( w ) = ∫ 0 ∞ Ψ ( x ) x w d x \widetilde{\Psi}(w)=\int_0^\infty\Psi(x)x^w\,dx Ψ ( w ) = ∫ 0 ∞ Ψ ( x ) x w d x p p p η v ( s ; l ) \eta_v(s;l) η v ( s ; l )
p p + 1 { ( 1 − p − s ) { 1 + p − 1 + p − s − ( p 2 v + p − 2 v ) p − s − 1 + p − 2 s − 1 } , p ∤ l , 1 − p − s , p ∣ l 1 , 1 − p − 2 s , p ∣ l , p ∤ l 1 , \frac{p}{p+1}
\begin{cases}
(1-p^{-s})\{1+p^{-1}+p^{-s}-(p^{2v}+p^{-2v})p^{-s-1}+p^{-2s-1}\}, & p\nmid l,\\
1-p^{-s}, & p\mid l_1,\\
1-p^{-2s}, & p\mid l,\ p\nmid l_1,
\end{cases} p + 1 p ⎩ ⎨ ⎧ ( 1 − p − s ) { 1 + p − 1 + p − s − ( p 2 v + p − 2 v ) p − s − 1 + p − 2 s − 1 } , 1 − p − s , 1 − p − 2 s , p ∤ l , p ∣ l 1 , p ∣ l , p ∤ l 1 , and the factor at 2 2 2 ( 1 − 2 − s − 2 v ) ( 1 − 2 − s ) ( 1 − 2 − s + 2 v ) (1-2^{-s-2v})(1-2^{-s})(1-2^{-s+2v}) ( 1 − 2 − s − 2 v ) ( 1 − 2 − s ) ( 1 − 2 − s + 2 v ) E c ( s ) = 1 E_c(s)=1 E c ( s ) = 1 c = 4 , 8 c=4,8 c = 4 , 8 E 1 ( s ) = E 2 ( s ) E_1(s)=E_2(s) E 1 ( s ) = E 2 ( s ) R 8 = 1 R_8=1 R 8 = 1 R 4 ( s , z ) = 2 − z R_4(s,z)=2^{-z} R 4 ( s , z ) = 2 − z R 1 = R ζ R_1=R_\zeta R 1 = R ζ d d d A δ , τ , c ( d ) = ( c d / π ) τ Γ a , δ ( τ ) L D ( 2 ) ( 1 / 2 + δ 1 ) L D ( 2 ) ( 1 / 2 + δ 2 ) A_{\delta,\tau,c}(d)=(cd/\pi)^\tau\Gamma_{a,\delta}(\tau)L_D^{(2)}(1/2+\delta_1)L_D^{(2)}(1/2+\delta_2) A δ , τ , c ( d ) = ( c d / π ) τ Γ a , δ ( τ ) L D ( 2 ) ( 1/2 + δ 1 ) L D ( 2 ) ( 1/2 + δ 2 ) H H H Y − 1 ∑ d ≡ r ( 8 ) ( ∑ α 2 ∣ d , α ≤ V μ ( α ) ) χ D ( l ) A δ , τ , c ( d ) Ψ ( d / Y ) Y^{-1}\sum_{d\equiv r(8)}(\sum_{\alpha^2\mid d,\ \alpha\le V}\mu(\alpha))\chi_D(l)A_{\delta,\tau,c}(d)\Psi(d/Y) Y − 1 ∑ d ≡ r ( 8 ) ( ∑ α 2 ∣ d , α ≤ V μ ( α )) χ D ( l ) A δ , τ , c ( d ) Ψ ( d / Y )
1 π 2 l 1 [ ∑ μ = ± 1 r δ ( l 1 ) Γ a , δ ( μ τ ) ( c Y π l 1 ) μ τ E c ( μ τ ) Ψ ~ ( μ τ ) Z ( 1 + 2 μ τ ; δ ) η δ ( 1 + 2 μ τ ; l ) + ∑ z = ± δ r τ ( l 1 ) Γ a , τ ( z ) ( 8 Y π l 1 ) z R c ( τ , z ) Ψ ~ ( z ) Z ( 1 + 2 z ; τ ) η τ ( 1 + 2 z ; l ) ] (75) \frac{1}{\pi^2\sqrt{l_1}}\left[
\sum_{\mu=\pm1}r_\delta(l_1)\Gamma_{a,\delta}(\mu\tau)\left(\frac{cY}{\pi l_1}\right)^{\mu\tau}E_c(\mu\tau)\widetilde{\Psi}(\mu\tau)Z(1+2\mu\tau;\delta)\eta_\delta(1+2\mu\tau;l)
+
\sum_{z=\pm\delta}r_\tau(l_1)\Gamma_{a,\tau}(z)\left(\frac{8Y}{\pi l_1}\right)^zR_c(\tau,z)\widetilde{\Psi}(z)Z(1+2z;\tau)\eta_\tau(1+2z;l)
\right] \tag*{(75)} π 2 l 1 1 [ μ = ± 1 ∑ r δ ( l 1 ) Γ a , δ ( μτ ) ( π l 1 c Y ) μτ E c ( μτ ) Ψ ( μτ ) Z ( 1 + 2 μτ ; δ ) η δ ( 1 + 2 μτ ; l ) + z = ± δ ∑ r τ ( l 1 ) Γ a , τ ( z ) ( π l 1 8 Y ) z R c ( τ , z ) Ψ ( z ) Z ( 1 + 2 z ; τ ) η τ ( 1 + 2 z ; l ) ] (75) with error
≪ ϵ Y ϵ { l 1 − 1 / 4 Y − 1 / 4 + l 1 − 1 / 2 V − 1 + l 1 / 2 V Y − 1 / 2 } . (76) \ll_\epsilon Y^\epsilon\{l_1^{-1/4}Y^{-1/4}+l_1^{-1/2}V^{-1}+l^{1/2}VY^{-1/2}\}. \tag*{(76)} ≪ ϵ Y ϵ { l 1 − 1/4 Y − 1/4 + l 1 − 1/2 V − 1 + l 1/2 V Y − 1/2 } . (76) Coincident residues in (75) mean their joint holomorphic continuation.
We justify both this continuation and the error bounds, rather than appealing to a change of conductor in the original calculation. The zero Fourier mode in (69) has l n ln l n n = l 1 m 2 n=l_1m^2 n = l 1 m 2 η \eta η ℜ s = − 1 / 4 + ϵ \Re s=-1/4+\epsilon ℜ s = − 1/4 + ϵ (76) and the residues at ± τ , ± δ \pm\tau,\pm\delta ± τ , ± δ (71) and (73) make the secondary integrand even. At s = z = ± δ s=z=\pm\delta s = z = ± δ R ζ ( z , z ) = 8 − z E 2 ( z ) R_\zeta(z,z)=8^{-z}E_2(z) R ζ ( z , z ) = 8 − z E 2 ( z ) (74) equals one there.
For nonprincipal branches, start the Fourier–Mellin calculation with w = − 1 / 2 + ϵ + i v w=-1/2+\epsilon+iv w = − 1/2 + ϵ + i v s s s b = ϵ b=\epsilon b = ϵ max ( ∣ ℜ τ ∣ , ℜ w ) < b < 1 + ℜ w \max(|\Re\tau|,\Re w)<b<1+\Re w max ( ∣ℜ τ ∣ , ℜ w ) < b < 1 + ℜ w b = ϵ + A 0 b = \epsilon+ A_{0} b = ϵ + A 0 Γ ( s − w ) \Gamma(s-w) Γ ( s − w ) Γ a , δ ( s ) \Gamma_{a,\delta}(s) Γ a , δ ( s ) E 2 ( s ) E_{2}(s) E 2 ( s ) s 2 − τ 2 s^{2}-\tau^{2} s 2 − τ 2
Y − 1 / 2 + ϵ α l ∣ j ∣ ( 1 + ∣ w ∣ ) C ∣ Ψ ~ ( w ) ∣ ( 1 + ∣ j ∣ ( α 2 l ) ) − A 0 . Y^{-1/2+\epsilon}\frac{\alpha\sqrt{l}}{\sqrt{|j|}}(1+|w|)^{C}|\widetilde{\Psi}(w)|\left(1+\frac{|j|}{(\alpha^{2}l)}\right)^{-A_{0}}. Y − 1/2 + ϵ ∣ j ∣ α l ( 1 + ∣ w ∣ ) C ∣ Ψ ( w ) ∣ ( 1 + ( α 2 l ) ∣ j ∣ ) − A 0 . The odd Euler series factors as two quadratic L L L
α ϵ ∣ j ∣ ϵ l 1 / 2 + ϵ ( l , j 2 2 ) 1 / 2 , \alpha^{\epsilon}|j|^{\epsilon}l^{1/2+\epsilon}(l,j_{2}^{2})^{1/2}, α ϵ ∣ j ∣ ϵ l 1/2 + ϵ ( l , j 2 2 ) 1/2 , where j = j 0 j 2 2 j=j_{0}j_{2}^{2} j = j 0 j 2 2 j 0 j_{0} j 0 j 2 j_{2} j 2 O ( α l / ∣ j 0 ∣ ) O(\alpha\sqrt{l/|j_{0}|}) O ( α l /∣ j 0 ∣ ) [10 Lemma 3.5] to the dyadic j 0 j_{0} j 0 ( α 2 l ) ϵ ( 1 + ∣ w ∣ ) C (\alpha^{2}l)^{\epsilon}(1+|w|)^{C} ( α 2 l ) ϵ ( 1 + ∣ w ∣ ) C α ≤ V \alpha\leq V α ≤ V (76) . The principal zeta factor, when present in the square-free tail, is handled with (5.14) before shifting contours.
For clarity, the shifts here have imaginary parts as large as L − 5 / 6 L^{-5/6} L − 5/6 O ( L − 1 ) O(L^{-1}) O ( L − 1 ) ϵ \epsilon ϵ 0 < ϑ < min ( 10 − 4 , ϵ / 100 ) 0<\vartheta<\min(10^{-4},\epsilon/100) 0 < ϑ < min ( 1 0 − 4 , ϵ /100 ) Y Y Y ϑ / 4 \vartheta/4 ϑ /4 α \alpha α ℜ s = ϑ + 1 / log Y \Re s=\vartheta+1/\log Y ℜ s = ϑ + 1/ log Y ∣ z ∣ = ϑ + 1 / ( 2 log Y ) |z|=\vartheta+1/(2\log Y) ∣ z ∣ = ϑ + 1/ ( 2 log Y ) 2 z 2z 2 z z = 0 z=0 z = 0 ℜ ( s − z ) ≥ 1 / ( 2 log Y ) \Re(s-z)\geq1/(2\log Y) ℜ ( s − z ) ≥ 1/ ( 2 log Y )
≪ ℓ 1 ϵ 1 l 1 2 ϑ − 1 / 2 Y ϑ + ϵ 1 V − 1 + 4 ϑ . \ll\ell_{1}^{\epsilon_{1}}l_{1}^{2\vartheta-1/2}Y^{\vartheta+\epsilon_{1}}V^{-1+4\vartheta}. ≪ ℓ 1 ϵ 1 l 1 2 ϑ − 1/2 Y ϑ + ϵ 1 V − 1 + 4 ϑ . With V = Y 1 / 20 V=Y^{1/20} V = Y 1/20 l ≤ M 2 ≤ Y .04 l\leq M^{2}\leq Y^{.04} l ≤ M 2 ≤ Y .04 ϵ 1 < ϵ / 10 \epsilon_{1}<\epsilon/10 ϵ 1 < ϵ /10 (76) . This Y ϵ Y^{\epsilon} Y ϵ (68) .
The full square-free remainder can be checked separately. Write d = l 2 m d=l^{2}m d = l 2 m m m m l > V l>V l > V Y l = Y / l 2 Y_{l}=Y/l^{2} Y l = Y / l 2 [10 Lemma 3.4] bounds the fourth mollifier sum by Y ϵ ( Y l + M 2 ) Y^{\epsilon}(Y_{l}+M^{2}) Y ϵ ( Y l + M 2 ) [10 Lemma 3.5] bound the second L L L Y ϵ Y l Y^{\epsilon}Y_{l} Y ϵ Y l m = 1 m=1 m = 1 M 2 M^{2} M 2 l l l Y ϵ ( V − 1 + M Y − 1 / 2 log Y ) Y^{\epsilon}(V^{-1}+MY^{-1/2}\log Y) Y ϵ ( V − 1 + M Y − 1/2 log Y ) (76) against the two mollifier coefficients therefore leaves at most
Y ϵ { M Y − 1 / 4 + M V − 1 + M 2 V Y − 1 / 2 + V − 1 + M Y − 1 / 2 } , (77) Y^{\epsilon}\{MY^{-1/4}+MV^{-1}+M^{2}VY^{-1/2}+V^{-1}+MY^{-1/2}\}, \tag*{(77)} Y ϵ { M Y − 1/4 + M V − 1 + M 2 V Y − 1/2 + V − 1 + M Y − 1/2 } , (77) a fixed negative power when .009 ≤ log M / log Y ≤ .02 .009\leq\log M/\log Y\leq.02 .009 ≤ log M / log Y ≤ .02
It remains to bound the main terms uniformly through colliding shifts. We give the cancellation estimate used for this purpose. For a generic term ( w , v ) (w,v) ( w , v ) (75) , its mollifier sum is
A ( w , v ) = ∑ r , s o d d λ ( r ) λ ( s ) r 1 / 2 + δ 1 s 1 / 2 + δ 2 r v ( l 1 ) l 1 1 / 2 + w η v ( 1 + 2 w ; r s ) , λ ( n ) = μ ( n ) Q ( log ( M / n ) / log M ) . A(w,v)=\sum_{\substack{r,s\ \mathrm{odd}}}\frac{\lambda(r)\lambda(s)}{r^{1/2+\delta_{1}}s^{1/2+\delta_{2}}}\frac{r_{v}(l_{1})}{l_{1}^{1/2+w}}\eta_{v}(1+2w;rs),\qquad\lambda(n)=\mu(n)Q(\log(M/n)/\log M). A ( w , v ) = r , s odd ∑ r 1/2 + δ 1 s 1/2 + δ 2 λ ( r ) λ ( s ) l 1 1/2 + w r v ( l 1 ) η v ( 1 + 2 w ; r s ) , λ ( n ) = μ ( n ) Q ( log ( M / n ) / log M ) . The square-free rearrangement of the Euler factors gives
A ( w , v ) = η v ( 1 + 2 w ; 1 ) ∑ γ o d d H w , v ( γ ) γ 1 + 2 τ h w , v ( γ ) S γ ( δ 1 + w , v ) S γ ( δ 2 + w , v ) , (78) A(w,v)=\eta_{v}(1+2w;1)\sum_{\gamma\ \mathrm{odd}}\frac{H_{w,v}(\gamma)}{\gamma^{1+2\tau h_{w,v}(\gamma)}}S_{\gamma}(\delta_{1}+w,v)S_{\gamma}(\delta_{2}+w,v), \tag*{(78)} A ( w , v ) = η v ( 1 + 2 w ; 1 ) γ odd ∑ γ 1 + 2 τ h w , v ( γ ) H w , v ( γ ) S γ ( δ 1 + w , v ) S γ ( δ 2 + w , v ) , (78) where
h w , v ( p ) = 1 + p − 1 + p − 1 − 2 w − ( p 2 v + p − 2 v ) p − 2 − 2 w + p − 3 − 4 w , H w , v ( p ) = 1 + p − 1 − 2 w − r v ( p ) 2 p − 1 − 2 w / h w , v ( p ) , S γ ( u , v ) = ∑ a ≤ M / γ a o d d r v ( a ) μ ( a ) a 1 + u h w , v ( a ) Q ( log ( M / ( a γ ) ) / log M ) . \begin{aligned}
h_{w,v}(p)&=1+p^{-1}+p^{-1-2w}-(p^{2v}+p^{-2v})p^{-2-2w}+p^{-3-4w},\\
H_{w,v}(p)&=1+p^{-1-2w}-r_{v}(p)^{2}p^{-1-2w}/h_{w,v}(p),\\
S_{\gamma}(u,v)&=\sum_{\substack{a\leq M/\gamma\\a\ \mathrm{odd}}}\frac{r_{v}(a)\mu(a)}{a^{1+u}h_{w,v}(a)}Q(\log(M/(a\gamma))/\log M).
\end{aligned} h w , v ( p ) H w , v ( p ) S γ ( u , v ) = 1 + p − 1 + p − 1 − 2 w − ( p 2 v + p − 2 v ) p − 2 − 2 w + p − 3 − 4 w , = 1 + p − 1 − 2 w − r v ( p ) 2 p − 1 − 2 w / h w , v ( p ) , = a ≤ M / γ a odd ∑ a 1 + u h w , v ( a ) r v ( a ) μ ( a ) Q ( log ( M / ( aγ )) / log M ) . Let E γ = ∏ p ∣ γ ( 1 + C / p ) E_\gamma=\prod_{p\mid\gamma}(1+C/\sqrt{p}) E γ = ∏ p ∣ γ ( 1 + C / p ) [10 Lemma 6.1] , applied to these explicit Euler factors, gives
∣ S γ ( u , v ) ∣ ≪ E γ H 6 ( ∣ u + v ∣ + L − 1 ) ( ∣ u − v ∣ + L − 1 ) . (79) |S_\gamma(u,v)|\ll E_\gamma H^6(|u+v|+L^{-1})(|u-v|+L^{-1}). \tag*{(79)} ∣ S γ ( u , v ) ∣ ≪ E γ H 6 ( ∣ u + v ∣ + L − 1 ) ( ∣ u − v ∣ + L − 1 ) . (79) Here are the estimates within that application. Its generating series is μ ( γ ) G ( 1 + s , γ ; u , v ) J ( s + u + v ) J ( s + u − v ) \mu(\gamma)G(1+s,\gamma;u,v)J(s+u+v)J(s+u-v) μ ( γ ) G ( 1 + s , γ ; u , v ) J ( s + u + v ) J ( s + u − v ) J ( z ) = 1 / ζ ( 1 + z ) = z + O ( z 2 ) J(z)=1/\zeta(1+z)=z+O(z^2) J ( z ) = 1/ ζ ( 1 + z ) = z + O ( z 2 ) G G G O ( E γ H 2 j ) O(E_\gamma H^{2j}) O ( E γ H 2 j ) 1 + O ( p − 1 ) 1+O(p^{-1}) 1 + O ( p − 1 ) ∑ p ∣ γ ( log p ) j / p ≪ 1 + H j + 1 \sum_{p\mid\gamma}(\log p)^j/p\ll1+H^{j+1} ∑ p ∣ γ ( log p ) j / p ≪ 1 + H j + 1 ( log M ) j (\log M)^j ( log M ) j O ( E γ / L 2 ) O(E_\gamma/L^2) O ( E γ / L 2 ) γ > M / 2 \gamma>M/2 γ > M /2 Q ( v ) = O ( v 2 ) Q(v)=O(v^2) Q ( v ) = O ( v 2 ) γ ≤ M 1 / 2 \gamma\le M^{1/2} γ ≤ M 1/2 M M M M 1 / 2 M^{1/2} M 1/2 (79) also across the join of Q Q Q
∑ γ ≤ M ∣ H w , v ( γ ) / h w , v ( γ ) ∣ E γ 2 γ ≪ ∏ p ≤ M ( 1 + p − 1 + C p − 3 / 2 ) ≪ L . \sum_{\gamma\le M}\frac{|H_{w,v}(\gamma)/h_{w,v}(\gamma)|E_\gamma^2}{\gamma}\ll\prod_{p\le M}(1+p^{-1}+Cp^{-3/2})\ll L. γ ≤ M ∑ γ ∣ H w , v ( γ ) / h w , v ( γ ) ∣ E γ 2 ≪ p ≤ M ∏ ( 1 + p − 1 + C p − 3/2 ) ≪ L . For δ 1 = i t \delta_1=it δ 1 = i t δ 2 = − i t \delta_2=-it δ 2 = − i t ∣ t ∣ ≥ 4 / L |t|\ge4/L ∣ t ∣ ≥ 4/ L ∣ w ∣ = 1 / L |w|=1/L ∣ w ∣ = 1/ L A ≪ H 12 ( ∣ t ∣ + L − 1 ) 2 / L A\ll H^{12}(|t|+L^{-1})^2/L A ≪ H 12 ( ∣ t ∣ + L − 1 ) 2 / L Z ( 1 + 2 w ; i t ) ≪ L / ( ∣ t ∣ + L − 1 ) 2 Z(1+2w;it)\ll L/(|t|+L^{-1})^2 Z ( 1 + 2 w ; i t ) ≪ L / ( ∣ t ∣ + L − 1 ) 2 2 / w 2/w 2/ w w = ± i t w=\pm it w = ± i t v = 0 v=0 v = 0 O ( H 12 / ( ∣ t ∣ L ) ) O(H^{12}/(|t|L)) O ( H 12 / ( ∣ t ∣ L )) O ( H 12 ) O(H^{12}) O ( H 12 ) ∣ t ∣ ≤ 4 / L |t|\le4/L ∣ t ∣ ≤ 4/ L ∣ δ 1 ∣ = 10 / L |\delta_1|=10/L ∣ δ 1 ∣ = 10/ L ∣ δ 2 ∣ = 20 / L |\delta_2|=20/L ∣ δ 2 ∣ = 20/ L Z = O ( L 3 ) Z=O(L^3) Z = O ( L 3 ) A = O ( H 12 L − 3 ) A=O(H^{12}L^{-3}) A = O ( H 12 L − 3 ) τ = 0 \tau=0 τ = 0
Divide by ( c Y / π ) τ Γ a , δ ( τ ) (cY/\pi)^\tau\Gamma_{a,\delta}(\tau) ( c Y / π ) τ Γ a , δ ( τ ) Ψ ( x ) = x − τ Φ ( x ) \Psi(x)=x^{-\tau}\Phi(x) Ψ ( x ) = x − τ Φ ( x ) Φ ^ ( 0 ) / π 2 + O ( Y − 1 / 2 ) \widehat{\Phi}(0)/\pi^2+O(Y^{-1/2}) Φ ( 0 ) / π 2 + O ( Y − 1/2 ) (77) –(79) prove the weighted bound, weakened to H 20 H^{20} H 20 [ 6 / 5 , 9 / 5 ] [6/5,9/5] [ 6/5 , 9/5 ] 5 Y / 6 5Y/6 5 Y /6 10 Y / 9 10Y/9 10 Y /9 ∣ D ∣ ≥ X .9 |D|\ge X^{.9} ∣ D ∣ ≥ X .9 M = X .01 M=X^{.01} M = X .01 .009 ≤ log M / log Y ≤ .02 .009\le\log M/\log Y\le.02 .009 ≤ log M / log Y ≤ .02 O ( X ) O(X) O ( X ) ∣ D ∣ < X .9 |D|<X^{.9} ∣ D ∣ < X .9 [10 Lemmas 3.4–3.5] give, by Cauchy–Schwarz, a normalized contribution O ( X − .08 ) O(X^{-.08}) O ( X − .08 ) (68) on the original family. □
Zero detectors in the three families
In this subsection a zero is written ρ = 1 / 2 + β + i γ \rho=1/2+\beta+i\gamma ρ = 1/2 + β + iγ N ( σ , T ) N(\sigma,T) N ( σ , T ) β > σ \beta>\sigma β > σ ∣ γ ∣ ≤ T |\gamma|\le T ∣ γ ∣ ≤ T
We first record the logarithmic fourth-moment bound needed to turn Proposition 5.4 into a zero detector:
E F ( X ) ∣ L ( 1 / 2 + i t , χ D ) ∣ 4 ≪ ( log X ) 10 ( 1 + ∣ t ∣ ) 8 . (80) \mathop{\mathbb{E}}_{\mathcal{F}(X)}|L(1/2+it,\chi_D)|^4\ll(\log X)^{10}(1+|t|)^8. \tag*{(80)} E F ( X ) ∣ L ( 1/2 + i t , χ D ) ∣ 4 ≪ ( log X ) 10 ( 1 + ∣ t ∣ ) 8 . (80) Here is a derivation with a harmless, but explicit, height exponent. For the fixed smooth partition function G G G [26 Lemma 2.9(ii)] , Proposition 1.3 of that paper bounds the dyadic conductor sum of ∣ ∑ χ D ( n ) n − 1 / 2 − i t G ( n / N ) ∣ 4 |\sum\chi_D(n)n^{-1/2-it}G(n/N)|^4 ∣ ∑ χ D ( n ) n − 1/2 − i t G ( n / N ) ∣ 4
C ( 1 + ∣ t ∣ ) 2 { M + N 2 log ( 2 + N 2 / M ) } log 6 ( 2 + M N ) . C(1+|t|)^2\{M+N^2\log(2+N^2/M)\}\log^6(2+MN). C ( 1 + ∣ t ∣ ) 2 { M + N 2 log ( 2 + N 2 / M )} log 6 ( 2 + M N ) . Put T = 1 + ∣ t ∣ T=1+|t| T = 1 + ∣ t ∣ N ≤ M T N\le\sqrt{MT} N ≤ M T O ( M T 4 log 6 ( 2 + M ) ) O(MT^4\log^6(2+M)) O ( M T 4 log 6 ( 2 + M )) log 7 ( 2 + T ) ≪ T \log^7(2+T)\ll T log 7 ( 2 + T ) ≪ T N > M T N>\sqrt{MT} N > M T N 1 = M T / N N_1=MT/N N 1 = M T / N
Γ ( 1 / 4 + ( s − i t + a ) / 2 ) Γ ( 1 / 4 − ( s − i t − a ) / 2 ) , a = 0 , 1. \frac{\Gamma(1/4+(s-it+a)/2)}{\Gamma(1/4-(s-it-a)/2)},\qquad a=0,1. Γ ( 1/4 − ( s − i t − a ) /2 ) Γ ( 1/4 + ( s − i t + a ) /2 ) , a = 0 , 1. Decompose the transformed series into dyadic blocks N 0 N_0 N 0 V ( x ) = G ( 2 x ) + G ( x ) + G ( x / 2 ) V(x)=G(2x)+G(x)+G(x/2) V ( x ) = G ( 2 x ) + G ( x ) + G ( x /2 ) G G G N 0 ≤ N 1 N_0\le N_1 N 0 ≤ N 1 ℜ s = − 1 / 8 \Re s=-1/8 ℜ s = − 1/8 − 1 / 2 -1/2 − 1/2 N 0 > N 1 N_0>N_1 N 0 > N 1 ℜ s = 4 \Re s=4 ℜ s = 4
C ∑ N 0 ≤ N 1 d y a d ( N 0 N 1 ) 1 / 8 ∫ R ∣ A D ( N 0 , v − t ) ∣ ( 1 + ∣ v ∣ ) 9 d v + C ∑ N 0 > N 1 d y a d ( N 1 N 0 ) 4 ∫ R ∣ A D ( N 0 , v − t ) ∣ ( 1 + ∣ v ∣ ) 9 d v . C\sum_{N_0\le N_1}^{\mathrm{dyad}} \left(\frac{N_0}{N_1}\right)^{1/8}\int_{\mathbb{R}}\frac{|A_D(N_0,v-t)|}{(1+|v|)^9}\,dv+C\sum_{N_0>N_1}^{\mathrm{dyad}} \left(\frac{N_1}{N_0}\right)^4\int_{\mathbb{R}}\frac{|A_D(N_0,v-t)|}{(1+|v|)^9}\,dv. C N 0 ≤ N 1 ∑ dyad ( N 1 N 0 ) 1/8 ∫ R ( 1 + ∣ v ∣ ) 9 ∣ A D ( N 0 , v − t ) ∣ d v + C N 0 > N 1 ∑ dyad ( N 0 N 1 ) 4 ∫ R ( 1 + ∣ v ∣ ) 9 ∣ A D ( N 0 , v − t ) ∣ d v . where A D ( N 0 , w ) = ∑ χ D ( n ) n − 1 / 2 − i w V ( n / N 0 ) A_D(N_0,w)=\sum\chi_D(n)n^{-1/2-iw}V(n/N_0) A D ( N 0 , w ) = ∑ χ D ( n ) n − 1/2 − i w V ( n / N 0 ) T ≤ ( 1 + ∣ v ∣ ) ( 1 + ∣ v − t ∣ ) T\le(1+|v|)(1+|v-t|) T ≤ ( 1 + ∣ v ∣ ) ( 1 + ∣ v − t ∣ ) ∣ A D ∣ ≪ N 0 |A_D|\ll\sqrt{N_0} ∣ A D ∣ ≪ N 0
C T 2 { ∑ N 0 ≤ N 1 ( N 0 N 1 ) 1 / 8 F M ( N 0 ) + ∑ N 0 > N 1 ( N 1 N 0 ) 4 F M ( N 0 ) } , CT^2\left\{\sum_{N_0\le N_1}\left(\frac{N_0}{N_1}\right)^{1/8}F_M(N_0)+\sum_{N_0>N_1}\left(\frac{N_1}{N_0}\right)^4F_M(N_0)\right\}, C T 2 { N 0 ≤ N 1 ∑ ( N 1 N 0 ) 1/8 F M ( N 0 ) + N 0 > N 1 ∑ ( N 0 N 1 ) 4 F M ( N 0 ) } , where F M ( v ) = { M + v 2 log ( 2 + v 2 / M ) } log 6 ( 2 + M v ) F_M(v)=\{M+v^2\log(2+v^2/M)\}\log^6(2+Mv) F M ( v ) = { M + v 2 log ( 2 + v 2 / M )} log 6 ( 2 + M v ) N 1 ≥ 1 N_1\ge1 N 1 ≥ 1 O ( T 2 F M ( N 1 ) ) O(T^2F_M(N_1)) O ( T 2 F M ( N 1 )) N 1 < 1 N_1<1 N 1 < 1 O ( M T 2 log 6 ( 2 + M ) ) O(MT^2\log^6(2+M)) O ( M T 2 log 6 ( 2 + M )) M T 4 log 6 ( 2 + M ) MT^4\log^6(2+M) M T 4 log 6 ( 2 + M ) N N N (62) bounds each smooth block in L 4 L^4 L 4 C L 3 / 2 T CL^{3/2}T C L 3/2 T Y ≍ X 4 T 4 Y\asymp X^4T^4 Y ≍ X 4 T 4 O ( X T Y − 1 / 2 ) = O ( 1 ) O(XTY^{-1/2})=O(1) O ( X T Y − 1/2 ) = O ( 1 ) O ( L + log ( 2 + T ) ) O(L+\log(2+T)) O ( L + log ( 2 + T )) (80) . This argument avoids moving a localized even-character gamma quotient past its pole at − 1 / 2 + i t -1/2+it − 1/2 + i t
Lemma 5.5. For the rational family let r = L / ℓ → ∞ r=L/\ell\to\infty r = L / ℓ → ∞ ℓ → ∞ \ell\to\infty ℓ → ∞ 0 ≤ T ∗ ≤ L − 5 / 6 / 32 0\le T_*\le L^{-5/6}/32 0 ≤ T ∗ ≤ L − 5/6 /32
1 − C H 20 ( 1 + T ∗ ℓ ) e − c r − C e − K H , (81) 1-CH^{20}(1+T_*\ell)e^{-cr}-Ce^{-KH}, \tag*{(81)} 1 − C H 20 ( 1 + T ∗ ℓ ) e − cr − C e − K H , (81) there is no zero with β > 2 / ℓ \beta>2/\ell β > 2/ ℓ ∣ γ ∣ ≤ T ∗ + ℓ − 3 ℓ β |\gamma|\le T_*+\ell^{-3\ell\beta} ∣ γ ∣ ≤ T ∗ + ℓ − 3 ℓ β K K K K K K ν = A H \nu=A\sqrt{H} ν = A H β > ν / ( 2 L ) \beta>\nu/(2L) β > ν / ( 2 L ) ∣ γ ∣ ≤ 2 β |\gamma|\le2\beta ∣ γ ∣ ≤ 2 β
≪ H 20 e − c ν + e − K H . (82) \ll H^{20}e^{-c\nu}+e^{-KH}. \tag*{(82)} ≪ H 20 e − c ν + e − K H . (82) There is also a coarse detector, valid for 8 ≤ ℓ ≤ c 0 L 8\le\ell\le c_0L 8 ≤ ℓ ≤ c 0 L α ≥ 1 / L \alpha\ge1/L α ≥ 1/ L T ∗ ≤ 1 T_*\le1 T ∗ ≤ 1 C L C G e − c G α L CL^CGe^{-cG\alpha L} C L C G e − c G α L β > α \beta>\alpha β > α
Proof. Put B D ( s ) = L ( s , χ D ) M D ( s ) − 1 B_D(s)=L(s,\chi_D)M_D(s)-1 B D ( s ) = L ( s , χ D ) M D ( s ) − 1 O ( L 6 ) O(L^6) O ( L 6 ) ∏ p ≤ M ( 1 + 6 / p + 1 / p 2 ) ≪ L 6 \prod_{p\le M}(1+6/p+1/p^2)\ll L^6 ∏ p ≤ M ( 1 + 6/ p + 1/ p 2 ) ≪ L 6 (61) is O ( X − 1 / 2 M 6 ) O(X^{-1/2}M^6) O ( X − 1/2 M 6 ) (80) give E ∣ B D ( 1 / 2 + i t ) ∣ 2 ≪ L 8 ( 1 + ∣ t ∣ ) C \mathbb{E}|B_D(1/2+it)|^2\ll L^8(1+|t|)^C E ∣ B D ( 1/2 + i t ) ∣ 2 ≪ L 8 ( 1 + ∣ t ∣ ) C 1 < n ≤ M 1 / 2 1<n\le M^{1/2} 1 < n ≤ M 1/2 L M − 1 LM-1 L M − 1 τ ( n ) \tau(n) τ ( n ) ∣ B D ( 2 + i t ) ∣ ≪ L X − 1 / 200 |B_D(2+it)|\ll LX^{-1/200} ∣ B D ( 2 + i t ) ∣ ≪ L X − 1/200 s + 3 s+3 s + 3 0 ≤ x ≤ 3 / 2 0\le x\le3/2 0 ≤ x ≤ 3/2
E ∣ B D ( 1 / 2 + x + i t ) ∣ 2 ≪ L 8 ( 1 + ∣ t ∣ ) C e − ( 2 / 300 ) x L . (83) \mathbb{E}|B_D(1/2+x+it)|^2\ll L^8(1+|t|)^C e^{-(2/300)xL}. \tag*{(83)} E ∣ B D ( 1/2 + x + i t ) ∣ 2 ≪ L 8 ( 1 + ∣ t ∣ ) C e − ( 2/300 ) xL . (83) At an L L L B D = − 1 B_D=-1 B D = − 1 1 / L 1/L 1/ L (83) give the coarse detector. More explicitly the area and height weights cost a fixed power of L L L e 3 ℓ β e^{3\ell\beta} e 3 ℓ β β L \beta L β L c 0 c_0 c 0 C L C G e − c G α L CL^CGe^{-cG\alpha L} C L C G e − c G α L
To remove the power of L L L h = L − 5 / 6 / 2 h=L^{-5/6}/2 h = L − 5/6 /2 b = B 0 H / L b=B_0H/L b = B 0 H / L u ( x , t ) = log ∥ B D ( 1 / 2 + x + i t ) ∥ L 2 ( F ( X ) ) u(x,t)=\log\|B_D(1/2+x+it)\|_{L^2(\mathcal{F}(X))} u ( x , t ) = log ∥ B D ( 1/2 + x + i t ) ∥ L 2 ( F ( X )) u ( 0 , t ) u(0,t) u ( 0 , t ) Q 0 = 1 2 log ( C ′ H 20 ) Q_0=\frac12\log(C'H^{20}) Q 0 = 2 1 log ( C ′ H 20 ) ∣ t ∣ ≤ h |t|\le h ∣ t ∣ ≤ h B 0 ≥ 16 C 0 / κ 0 B_0\ge16C_0/\kappa_0 B 0 ≥ 16 C 0 / κ 0 u ≤ C 0 H − κ 0 x L u\le C_0H-\kappa_0xL u ≤ C 0 H − κ 0 xL (83) . The subharmonic function u − Q 0 + ( κ 0 / 2 ) L x u-Q_0+(\kappa_0/2)Lx u − Q 0 + ( κ 0 /2 ) Lx ( 0 , b ) × ( − h , h ) (0,b)\times(-h,h) ( 0 , b ) × ( − h , h ) C 0 H C_0H C 0 H
p ( x , t ) = ∑ n o d d 4 n π sin ( n π x / b ) cosh ( n π t / b ) cosh ( n π h / b ) ≤ 9 x b e − π h / ( 2 b ) ( ∣ t ∣ ≤ h / 2 ) . p(x,t)=\sum_{n\ {\rm odd}}\frac{4}{n\pi}\sin(n\pi x/b)\frac{\cosh(n\pi t/b)}{\cosh(n\pi h/b)}\leq\frac{9x}{b}e^{-\pi h/(2b)}\qquad(|t|\leq h/2). p ( x , t ) = n odd ∑ nπ 4 sin ( nπ x / b ) cosh ( nπ h / b ) cosh ( nπ t / b ) ≤ b 9 x e − π h / ( 2 b ) ( ∣ t ∣ ≤ h /2 ) . Since h / b → ∞ h/b\to\infty h / b → ∞
E ∣ B D ( 1 / 2 + x + i t ) ∣ 2 ≪ H 20 e − κ 0 x L / 2 ( 0 ≤ x ≤ b , ∣ t ∣ ≤ h / 2 ) . (84) \mathbb{E}|B_D(1/2+x+it)|^2\ll H^{20}e^{-\kappa_0xL/2}\qquad(0\leq x\leq b,|t|\leq h/2). \tag*{(84)} E ∣ B D ( 1/2 + x + i t ) ∣ 2 ≪ H 20 e − κ 0 xL /2 ( 0 ≤ x ≤ b , ∣ t ∣ ≤ h /2 ) . (84) Submean disks of radius σ / 4 \sigma/4 σ /4 σ < β ≤ 2 σ \sigma<\beta\leq2\sigma σ < β ≤ 2 σ ∣ γ ∣ ≤ T |\gamma|\leq T ∣ γ ∣ ≤ T 3 σ / 4 ≤ x ≤ 9 σ / 4 3\sigma/4\leq x\leq9\sigma/4 3 σ /4 ≤ x ≤ 9 σ /4 ∣ t ∣ ≤ T + σ / 4 |t|\leq T+\sigma/4 ∣ t ∣ ≤ T + σ /4 (84) gives probability
≪ H 20 ( 1 + T / σ ) e − c σ L (85) \ll H^{20}(1+T/\sigma)e^{-c\sigma L} \tag*{(85)} ≪ H 20 ( 1 + T / σ ) e − c σ L (85) provided this rectangle lies in the preceding one.
Take a c = A c H / L a_c=A_cH/L a c = A c H / L A c = ( K + C G + 2 ) / c G A_c=(K+C_G+2)/c_G A c = ( K + C G + 2 ) / c G B 0 B_0 B 0 8 A c 8A_c 8 A c β > a c \beta>a_c β > a c O ( e − K H ) O(e^{-KH}) O ( e − K H ) a c a_c a c j / ℓ < β ≤ ( j + 1 ) / ℓ j/\ell<\beta\leq(j+1)/\ell j / ℓ < β ≤ ( j + 1 ) / ℓ j ≥ 2 j\geq2 j ≥ 2 2 ( T ∗ + ℓ − 1 ξ 3 ( j + 1 ) ) 2(T_*+\ell^{-1}\xi^{3(j+1)}) 2 ( T ∗ + ℓ − 1 ξ 3 ( j + 1 ) ) (85) . If 2 / ℓ ≥ a c 2/\ell\geq a_c 2/ ℓ ≥ a c − H + log r + 3 A c H / r + O ( 1 ) = − H + o ( H ) -H+\log r+3A_cH/r+O(1)=-H+o(H) − H + log r + 3 A c H / r + O ( 1 ) = − H + o ( H ) o ( h ) o(h) o ( h ) C H 20 ( 1 + T ∗ ℓ ) ∑ j ≥ 2 ( 1 + e 3 ( j + 1 ) ) e − c r j CH^{20}(1+T_*\ell)\sum_{j\geq2}(1+e^{3(j+1)})e^{-crj} C H 20 ( 1 + T ∗ ℓ ) ∑ j ≥ 2 ( 1 + e 3 ( j + 1 ) ) e − cr j (81) . For the wedge use the same high-zero cutoff and, below it, the bands σ = 2 j − 1 ν / L \sigma=2^{j-1}\nu/L σ = 2 j − 1 ν / L T = 4 σ T=4\sigma T = 4 σ (82) . □
We next derive the Hecke-family detector from the precise range of Gao’s mollified moment.
Fix κ = 10 − 3 \kappa=10^{-3} κ = 1 0 − 3 m = 1 / 2 − 5 κ m=1/2-5\kappa m = 1/2 − 5 κ b = 1 / 2 b=1/2 b = 1/2 M = X m M=X^m M = X m Q Q Q b b b n ≡ 1 ( m o d ( 1 + i ) 3 ) n\equiv1\pmod{(1+i)^3} n ≡ 1 ( mod ( 1 + i ) 3 ) μ K ( n ) Q ( log ( M / N n ) / log M ) \mu_K(n)Q(\log(M/Nn)/\log M) μ K ( n ) Q ( log ( M / N n ) / log M ) ( N n ) s (Nn)^s ( N n ) s Φ \Phi Φ [ 1 , 2 ] [1,2] [ 1 , 2 ] W ( z ) = E Φ ∣ L d ( 1 / 2 + z ) M d ( 1 / 2 + z ) ∣ 2 W(z)=\mathbb{E}_{\Phi}|L_d(1/2+z)M_d(1/2+z)|^2 W ( z ) = E Φ ∣ L d ( 1/2 + z ) M d ( 1/2 + z ) ∣ 2 [14 ] states, with z = τ + i t z=\tau+it z = τ + i t L 0 = log ( 32 X / π 2 ) L_0=\log(32X/\pi^2) L 0 = log ( 32 X / π 2 )
W ( z ) = 1 + { 1 − e − 2 τ L 0 2 τ m L − e − τ L 0 sin ( t L 0 ) t m L } ∫ 0 b e − 2 τ m L ( 1 − v ) ∣ Q ′ ( v ) + Q ′ ′ ( v ) 2 z m L ∣ 2 d v + O Φ ( X − κ − ϵ + e − 2 m ( 1 − b ) τ L ∣ z ∣ 6 L 5 ) . (86) \begin{aligned}
W(z)=1+\left\{\frac{1-e^{-2\tau L_0}}{2\tau mL}-e^{-\tau L_0}\frac{\sin(tL_0)}{tmL}\right\}\int_0^b e^{-2\tau mL(1-v)}\left|Q'(v)+\frac{Q''(v)}{2zmL}\right|^2\,dv\\
+O_\Phi\left(X^{-\kappa-\epsilon}+e^{-2m(1-b)\tau L}|z|^6L^5\right). \tag*{(86)}
\end{aligned} W ( z ) = 1 + { 2 τ m L 1 − e − 2 τ L 0 − e − τ L 0 t m L sin ( t L 0 ) } ∫ 0 b e − 2 τ m L ( 1 − v ) Q ′ ( v ) + 2 z m L Q ′′ ( v ) 2 d v + O Φ ( X − κ − ϵ + e − 2 m ( 1 − b ) τ L ∣ z ∣ 6 L 5 ) . (86) for fixed ϵ > 0 \epsilon>0 ϵ > 0 τ ≥ − 1 / ( ϵ L ) \tau\geq-1/(\epsilon L) τ ≥ − 1/ ( ϵ L ) ϵ / L ≤ ∣ z ∣ ≤ κ \epsilon/L\leq|z|\leq\kappa ϵ / L ≤ ∣ z ∣ ≤ κ L M LM L M L M − 1 LM-1 L M − 1 η 0 = m ( 1 − b ) \eta_0=m(1-b) η 0 = m ( 1 − b ) τ ≥ 1 / L \tau\geq1/L τ ≥ 1/ L sinh y / y ≥ 1 \sinh y/y\geq1 sinh y / y ≥ 1 sin y / y ≤ 1 \sin y/y\leq1 sin y / y ≤ 1 ∣ z ∣ L ≥ 1 |z|L\geq1 ∣ z ∣ L ≥ 1 O ( e − 2 η 0 τ L ) O(e^{-2\eta_0\tau L}) O ( e − 2 η 0 τ L ) τ 6 L 5 \tau^6L^5 τ 6 L 5
∣ W ( τ + i t ) − 1 ∣ ≪ ( 1 + ∣ t ∣ 6 L 5 ) e − η 0 τ L + X − κ . (87) |W(\tau+it)-1|\ll(1+|t|^6L^5)e^{-\eta_0\tau L}+X^{-\kappa}. \tag*{(87)} ∣ W ( τ + i t ) − 1∣ ≪ ( 1 + ∣ t ∣ 6 L 5 ) e − η 0 τ L + X − κ . (87) In particular the loss is logarithm-free only at height ∣ t ∣ ≤ L − 5 / 6 |t|\leq L^{-5/6} ∣ t ∣ ≤ L − 5/6
Lemma 5.6. There are fixed positive σ 0 , τ 0 , c \sigma_0,\tau_0,c σ 0 , τ 0 , c
E D X N ( σ , T ) ≪ ( 1 + T L ) ( 1 + T 6 L 5 ) e − c σ L ( 4 / L ≤ σ ≤ σ 0 , 0 ≤ T ≤ τ 0 ) . (88) \mathbb{E}_{D_X}N(\sigma,T)\ll(1+TL)(1+T^6L^5)e^{-c\sigma L}\qquad(4/L\leq\sigma\leq\sigma_0,\ 0\leq T\leq\tau_0). \tag*{(88)} E D X N ( σ , T ) ≪ ( 1 + T L ) ( 1 + T 6 L 5 ) e − c σ L ( 4/ L ≤ σ ≤ σ 0 , 0 ≤ T ≤ τ 0 ) . (88) For every fixed α > 0 \alpha>0 α > 0 θ α > 0 \theta_\alpha>0 θ α > 0
P D X { N ( α , X θ α ) > 0 } ≪ X − c α . (89) \mathbb{P}_{D_X}\{N(\alpha,X^{\theta_\alpha})>0\}\ll X^{-c\alpha}. \tag*{(89)} P D X { N ( α , X θ α ) > 0 } ≪ X − c α . (89) Also, for 8 ≤ ν = o ( L ) 8\leq\nu=o(L) 8 ≤ ν = o ( L ) β > ν / ( 2 L ) \beta>\nu/(2L) β > ν / ( 2 L ) ∣ γ ∣ ≤ 2 β |\gamma|\leq2\beta ∣ γ ∣ ≤ 2 β
≪ ν e − c ν + X − c 0 . (90) \ll\nu e^{-c\nu}+X^{-c_0}. \tag*{(90)} ≪ ν e − c ν + X − c 0 . (90) Proof. We spell out the application of Selberg’s rectangle lemma, because discarding its signed logarithm would lose (88) . Take λ = η 0 / 100 \lambda=\eta_0/100 λ = η 0 /100 S = π / ( 2 λ ) S=\pi/(2\lambda) S = π / ( 2 λ ) h = S / L h=S/L h = S / L 1 / 2 + σ / 2 1/2+\sigma/2 1/2 + σ /2 2 2 2 β > σ \beta>\sigma β > σ ∣ γ − t 0 ∣ ≤ h / 2 \lvert\gamma-t_0\rvert\le h/2 ∣ γ − t 0 ∣ ≤ h /2 4 h cos ( π / 4 ) sinh ( λ σ L / 2 ) 4h\cos(\pi/4)\sinh(\lambda\sigma L/2) 4 h cos ( π /4 ) sinh ( λσ L /2 ) log ∣ L M ∣ \log\lvert LM\rvert log ∣ L M ∣ E Φ log ∣ L M ∣ ≤ 1 2 log W ≤ 1 2 ∣ W − 1 ∣ \mathbb{E}_{\Phi}\log\lvert LM\rvert\le\frac{1}{2}\log W\le\frac{1}{2}\lvert W-1\rvert E Φ log ∣ L M ∣ ≤ 2 1 log W ≤ 2 1 ∣ W − 1 ∣ C h { ( 1 + ( T + 2 h ) ℓ L 5 ) e − η 0 σ L / 2 + X − κ } Ch\{(1+(T+2h)^{\ell}L^5)e^{-\eta_0\sigma L/2}+X^{-\kappa}\} C h {( 1 + ( T + 2 h ) ℓ L 5 ) e − η 0 σ L /2 + X − κ } u u u [ σ / 2 , κ / 2 ] [\sigma/2,\kappa/2] [ σ /2 , κ /2 ] [ κ / 2 , .6 ] [\kappa/2,.6] [ κ /2 , .6 ] [ .6 , 1.5 ] [.6,1.5] [ .6 , 1.5 ] (87) and integrates to C ( 1 + ( T + 2 h ) ℓ L 5 ) L − r C(1+(T+2h)^{\ell}L^5)L^{-r} C ( 1 + ( T + 2 h ) ℓ L 5 ) L − r ϵ 0 < η 0 κ / 100 \epsilon_0<\eta_0\kappa/100 ϵ 0 < η 0 κ /100
∣ W ( u + i t ) − 1 ∣ ≪ X ϵ 0 { e − 2 η 0 u L + e ( η 0 ( 1 / 2 − u ) − 1 / 2 ) L } . \lvert W(u+it)-1\rvert\ll X^{\epsilon_0}\{e^{-2\eta_0uL}+e^{(\eta_0(1/2-u)-1/2)L}\}. ∣ W ( u + i t ) − 1 ∣ ≪ X ϵ 0 { e − 2 η 0 uL + e ( η 0 ( 1/2 − u ) − 1/2 ) L } . After multiplication by the boundary weight e λ u L e^{\lambda uL} e λ uL L L L ∣ L M ( 1 / 2 + u + i t ) − 1 ∣ ≪ L e − η 0 ( u − 1 / 2 ) L \lvert LM(1/2+u+it)-1\rvert\ll Le^{-\eta_0(u-1/2)L} ∣ L M ( 1/2 + u + i t ) − 1 ∣ ≪ L e − η 0 ( u − 1/2 ) L L M LM L M ∣ log L M ( 2 + i t ) ∣ ≪ L e − η 0 L \lvert\log LM(2+it)\rvert\ll Le^{-\eta_0L} ∣ log L M ( 2 + i t )∣ ≪ L e − η 0 L X X X [ − T , T ] [-T,T] [ − T , T ] O ( 1 + T L ) O(1+TL) O ( 1 + T L ) σ 0 , τ 0 \sigma_0,\tau_0 σ 0 , τ 0 (88) . Two fixed plateaus at scales 5 X / 6 5X/6 5 X /6 10 X / 9 10X/9 10 X /9 4 / L 4/L 4/ L ∣ D X ∣ \lvert D_X\rvert ∣ D X ∣
For (89) a polynomial height bound is needed, rather than the compact-height statement of Proposition 3.2 alone. Its proof, using Gao’s Lemmas 2.7–2.8, decomposes B = L M − 1 B=LM-1 B = L M − 1 T 1 − T 2 T_1-T_2 T 1 − T 2 E ∣ T 1 ∣ 2 ≪ X ϵ − 2 η 0 u L \mathbb{E}\lvert T_1\rvert^2\ll X^{\epsilon-2\eta_0uL} E ∣ T 1 ∣ 2 ≪ X ϵ − 2 η 0 uL n − i t n^{-it} n − i t ℜ w = − u \Re w=-u ℜ w = − u e − N n / X e^{-Nn/X} e − N n / X α / 2 ≤ u ≤ .7 \alpha/2\le u\le.7 α /2 ≤ u ≤ .7
E ∣ L ( 1 / 2 + i v ) ∣ 4 ≪ X ϵ ( 1 + v 2 ) 1 + ϵ , E ∣ M ( 1 / 2 + i v ) ∣ 4 ≪ X ϵ ( 2 + M 2 / X ) \mathbb{E}\lvert L(1/2+iv)\rvert^4\ll X^\epsilon(1+v^2)^{1+\epsilon},\qquad\mathbb{E}\lvert M(1/2+iv)\rvert^4\ll X^\epsilon(2+M^2/X) E ∣ L ( 1/2 + i v ) ∣ 4 ≪ X ϵ ( 1 + v 2 ) 1 + ϵ , E ∣ M ( 1/2 + i v ) ∣ 4 ≪ X ϵ ( 2 + M 2 / X ) and gamma decay imply E ∣ T 2 ∣ 2 ≪ X − 2 u + ϵ ( 1 + ∣ t ∣ ) 4 \mathbb{E}\lvert T_2\rvert^2\ll X^{-2u+\epsilon}(1+\lvert t\rvert)^4 E ∣ T 2 ∣ 2 ≪ X − 2 u + ϵ ( 1 + ∣ t ∣ ) 4 E ∣ B ( 1 / 2 + u + i t ) ∣ 2 ≪ X ϵ ( 1 + ∣ t ∣ ) 4 e − 2 η 0 u L \mathbb{E}\lvert B(1/2+u+it)\rvert^2\ll X^\epsilon(1+\lvert t\rvert)^4e^{-2\eta_0uL} E ∣ B ( 1/2 + u + i t ) ∣ 2 ≪ X ϵ ( 1 + ∣ t ∣ ) 4 e − 2 η 0 uL β ≥ α \beta\ge\alpha β ≥ α ∣ γ ∣ ≤ T \lvert\gamma\rvert\le T ∣ γ ∣ ≤ T O α ( T + 1 ) O_\alpha(T+1) O α ( T + 1 ) α / 8 \alpha/8 α /8 B = − 1 B=-1 B = − 1 L L L C α X ϵ − η 0 α ( 1 + T ) 5 C_\alpha X^{\epsilon-\eta_0\alpha}(1+T)^5 C α X ϵ − η 0 α ( 1 + T ) 5 ϵ = θ α = η 0 α / 100 \epsilon=\theta_\alpha=\eta_0\alpha/100 ϵ = θ α = η 0 α /100 u u u (89) .
Finally take β 0 < min ( σ 0 / 2 , τ 0 / 16 ) \beta_0<\min(\sigma_0/2,\tau_0/16) β 0 < min ( σ 0 /2 , τ 0 /16 ) β > β 0 \beta>\beta_0 β > β 0 (89) . Below it use the bands b j = 2 j − 1 ν / L b_j=2^{j-1}\nu/L b j = 2 j − 1 ν / L 8 b j 8b_j 8 b j (88) . Writing v j = b j L v_j=b_jL v j = b j L C v j ( 1 + v j 6 / L ) e − c v j ≪ C v j e − c v j / 2 Cv_j(1+v_j^6/L)e^{-cv_j}\ll Cv_je^{-cv_j/2} C v j ( 1 + v j 6 / L ) e − c v j ≪ C v j e − c v j /2 (90) .
For the weight family the needed zero density is a direct literature input. Hough’s Main Theorem 1.1 [15 ] states, for some sufficiently small fixed positive θ , δ \theta,\delta θ , δ
E H k N ( σ , T ) ≪ T log k e − θ σ log k , 2 log k < σ < 1 2 , 10 log k < T < k δ . (91) \mathbb{E}H_kN(\sigma,T)\ll T\log k\,e^{-\theta\sigma\log k},\qquad\frac{2}{\log k}<\sigma<\frac{1}{2},\qquad\frac{10}{\log k}<T<k^\delta. \tag*{(91)} E H k N ( σ , T ) ≪ T log k e − θ σ l o g k , log k 2 < σ < 2 1 , log k 10 < T < k δ . (91) Its measure is unweighted and its range includes all even weights. Set L = log k L=\log k L = log k β \beta β 1 / ℓ 1/\ell 1/ ℓ r = L / ℓ → ∞ r=L/\ell\to\infty r = L / ℓ → ∞ ℓ → ∞ \ell\to\infty ℓ → ∞ T ∗ ≤ 1 T_*\le1 T ∗ ≤ 1
P { ∃ ρ : β > 2 / ℓ , ∣ γ ∣ ≤ T ∗ + ℓ − 1 e 3 ℓ β } ≪ ( T ∗ L + r ) e − c r . (92) \mathbb{P}\{\exists\rho:\beta>2/\ell,\lvert\gamma\rvert\le T_*+\ell^{-1}e^{3\ell\beta}\}\ll(T_*L+r)e^{-cr}. \tag*{(92)} P { ∃ ρ : β > 2/ ℓ , ∣ γ ∣ ≤ T ∗ + ℓ − 1 e 3 ℓ β } ≪ ( T ∗ L + r ) e − cr . (92) Indeed the j j j T j = 2 ( T ∗ + ℓ − 1 e 3 ( j + 1 ) ) T_j=2(T_*+\ell^{-1}e^{3(j+1)}) T j = 2 ( T ∗ + ℓ − 1 e 3 ( j + 1 ) ) j ≥ 2 j\ge2 j ≥ 2 10 / L 10/L 10/ L 2 / L 2/L 2/ L e 3 ℓ / 2 + O ( 1 ) = k O ( 1 ) e^{3\ell/2+O(1)}=k^{O(1)} e 3 ℓ /2 + O ( 1 ) = k O ( 1 ) (91) , and summing C ( T ∗ L + r e 3 ( j + 1 ) ) e − c r j C(T_*L+re^{3(j+1)})e^{-crj} C ( T ∗ L + r e 3 ( j + 1 ) ) e − cr j 2 j − 1 ν / L 2^{j-1}\nu/L 2 j − 1 ν / L
P { ∃ ρ : β > ν / ( 2 L ) , ∣ γ ∣ ≤ 2 β } ≪ ν e − c ν , ν → ∞ , ν = o ( L ) . (93) \mathbb{P}\{\exists\rho:\beta>\nu/(2L),\lvert\gamma\rvert\le2\beta\}\ll\nu e^{-c\nu},\qquad\nu\to\infty,\quad\nu=o(L). \tag*{(93)} P { ∃ ρ : β > ν / ( 2 L ) , ∣ γ ∣ ≤ 2 β } ≪ ν e − c ν , ν → ∞ , ν = o ( L ) . (93) A zero-free complex approximation
Write L = − L ′ / L \mathcal{L}=-L'/L L = − L ′ / L
K ℓ ( z ) = e ℓ z ( e ℓ z − 1 ) 2 ℓ 2 z 3 , P ℓ ( s ) = ∑ n ≤ e 3 ℓ Λ L ( n ) w ℓ ( n ) n − s , K_{\ell}(z)=\frac{e^{\ell z}(e^{\ell z}-1)^2}{\ell^2z^3},\qquad P_{\ell}(s)=\sum_{n\leq e^{3\ell}}\Lambda_L(n)w_{\ell}(n)n^{-s}, K ℓ ( z ) = ℓ 2 z 3 e ℓ z ( e ℓ z − 1 ) 2 , P ℓ ( s ) = n ≤ e 3 ℓ ∑ Λ L ( n ) w ℓ ( n ) n − s , where Λ L ( p j ) = χ D ( p j ) log p \Lambda_L(p^j)=\chi_D(p^j)\log p Λ L ( p j ) = χ D ( p j ) log p Λ L ( p j ) = A j ( λ f ( p ) ) log p \Lambda_L(p^j)=A_j(\lambda_f(p))\log p Λ L ( p j ) = A j ( λ f ( p )) log p
w ℓ ( n ) = ( 3 ℓ − log n ) + 2 − 2 ( 2 ℓ − log n ) + 2 + ( ℓ − log n ) + 2 2 ℓ 2 . w_{\ell}(n)=\frac{(3\ell-\log n)_+^2-2(2\ell-\log n)_+^2+(\ell-\log n)_+^2}{2\ell^2}. w ℓ ( n ) = 2 ℓ 2 ( 3 ℓ − log n ) + 2 − 2 ( 2 ℓ − log n ) + 2 + ( ℓ − log n ) + 2 . The weight lies in [ 0 , 1 ] [0,1] [ 0 , 1 ] n ≤ e ℓ n\leq e^\ell n ≤ e ℓ n > e 3 ℓ n>e^{3\ell} n > e 3 ℓ P ℓ P_\ell P ℓ C e 9 ℓ / 2 Ce^{9\ell/2} C e 9 ℓ /2 C e 7 ℓ Ce^{7\ell} C e 7 ℓ A j A_j A j e p j − e p j − 2 e^{p_j}-e^{p_{j-2}} e p j − e p j − 2 τ ( n ) ≤ 2 n \tau(n)\leq2\sqrt{n} τ ( n ) ≤ 2 n C ∑ n ≤ e 3 ℓ n log n C\sum_{n\leq e^{3\ell}}n\log n C ∑ n ≤ e 3 ℓ n log n
The Gaussian Hecke family needs a kernel with stronger vertical decay. Choose a fixed τ > 0 \tau>0 τ > 0 τ 0 / 10 \tau_0/10 τ 0 /10 a = 4 / τ a=4/\tau a = 4/ τ k ℓ = ⌊ ℓ / a ⌋ k_\ell=\lfloor\ell/a\rfloor k ℓ = ⌊ ℓ / a ⌋
K ℓ ∗ ( z ) = K ℓ ( z ) ( sinh ( a z / 2 ) a z / 2 ) k ℓ . (94) K_\ell^*(z)=K_\ell(z)\left(\frac{\sinh(az/2)}{az/2}\right)^{k_\ell}. \tag*{(94)} K ℓ ∗ ( z ) = K ℓ ( z ) ( a z /2 sinh ( a z /2 ) ) k ℓ . (94) If U 1 , U 2 U_1,U_2 U 1 , U 2 [ 0 , ℓ ] [0,\ell] [ 0 , ℓ ] V j V_j V j [ − a / 2 , a / 2 ] [-a/2,a/2] [ − a /2 , a /2 ] K ℓ ∗ ( z ) = E e V z / z K_\ell^*(z)=\mathbb{E}e^{Vz}/z K ℓ ∗ ( z ) = E e V z / z V = ℓ + U 1 + U 2 + ∑ V j ∈ [ ℓ / 2 , 7 ℓ / 2 ] V=\ell+U_1+U_2+\sum V_j\in[\ell/2,7\ell/2] V = ℓ + U 1 + U 2 + ∑ V j ∈ [ ℓ /2 , 7 ℓ /2 ] w ℓ ∗ ( n ) = P ( V ≥ log n ) w_\ell^*(n)=\mathbb{P}(V\geq\log n) w ℓ ∗ ( n ) = P ( V ≥ log n ) e ℓ / 2 e^{\ell/2} e ℓ /2 e 7 ℓ / 2 e^{7\ell/2} e 7 ℓ /2 P ℓ ∗ P_\ell^* P ℓ ∗ C e 8 ℓ Ce^{8\ell} C e 8 ℓ
∣ K ℓ ∗ ( u + i v ) ∣ ≤ 4 ℓ 2 ∣ u + i v ∣ 3 { e 7 ℓ u / 2 , u ≥ 0 , e ℓ u / 2 , u ≤ 0. (95) |K_\ell^*(u+iv)|\leq\frac{4}{\ell^2|u+iv|^3}
\begin{cases}
e^{7\ell u/2}, & u\geq0,\\
e^{\ell u/2}, & u\leq0.
\end{cases} \tag*{(95)} ∣ K ℓ ∗ ( u + i v ) ∣ ≤ ℓ 2 ∣ u + i v ∣ 3 4 { e 7 ℓ u /2 , e ℓ u /2 , u ≥ 0 , u ≤ 0. (95) Moreover, for β ≤ β 0 = τ / 1000 \beta\leq\beta_0=\tau/1000 β ≤ β 0 = τ /1000 δ ≥ 0 \delta\geq0 δ ≥ 0 ∣ v ∣ ≥ τ |v|\geq\tau ∣ v ∣ ≥ τ
∣ K ℓ ∗ ( β − δ + i v ) ∣ ≤ 4 e − ℓ δ / 2 − τ ℓ / 20 ℓ 2 ∣ v ∣ 3 ( τ ∣ v ∣ ) k ℓ . (96) |K_\ell^*(\beta-\delta+iv)|\leq\frac{4e^{-\ell\delta/2-\tau\ell/20}}{\ell^2|v|^3}\left(\frac{\tau}{|v|}\right)^{k_\ell}. \tag*{(96)} ∣ K ℓ ∗ ( β − δ + i v ) ∣ ≤ ℓ 2 ∣ v ∣ 3 4 e − ℓ δ /2 − τ ℓ /20 ( ∣ v ∣ τ ) k ℓ . (96) Indeed the extra factor is at most { 1 2 e a ∣ u ∣ / 2 τ / ∣ v ∣ } k ℓ \{ \frac{1}{2}e^{a|u|/2}\tau/|v|\}^{k_\ell} { 2 1 e a ∣ u ∣/2 τ /∣ v ∣ } k ℓ k ℓ ≥ τ ℓ / 8 k_\ell\geq\tau\ell/8 k ℓ ≥ τ ℓ /8 ℓ \ell ℓ 7 β 0 ℓ / 2 7\beta_0\ell/2 7 β 0 ℓ /2 2 − k ℓ 2^{-k_\ell} 2 − k ℓ
We use the following basic analytic facts in their unconditional form. The primitive completed functions are entire of order one, their zeros are in 0 ≤ ℜ s ≤ 1 0\leq\Re s\leq1 0 ≤ ℜ s ≤ 1 ρ \rho ρ 1 − ρ 1-\rho 1 − ρ ρ ˉ \bar{\rho} ρ ˉ 2 + i T 2+iT 2 + i T
# { ρ : ∣ ℑ ρ − T ∣ ≤ 1 } ≪ log ( Q ( ∣ T ∣ + 2 ) ) (97) \#\{\rho:|\Im\rho-T|\leq1\}\ll\log(Q(|T|+2)) \tag*{(97)} # { ρ : ∣ℑ ρ − T ∣ ≤ 1 } ≪ log ( Q ( ∣ T ∣ + 2 )) (97) in the degree-one families. The lower bound at the centre of the Jensen disk is supplied by the absolutely convergent Euler product. For weight k k k O ( log ( k + ∣ T ∣ + 2 ) ) O(\log(k+|T|+2)) O ( log ( k + ∣ T ∣ + 2 )) 2 + i T 2+iT 2 + i T 1 / 5 1/5 1/5
Writing z = s − 1 / 2 z=s-1/2 z = s − 1/2
L ( 1 / 2 + z ) = B ( z ) − m 0 z − ∑ α ∈ P 2 z z 2 − α 2 . (98) \mathcal{L}(1/2+z)=B(z)-\frac{m_0}{z}-\sum_{\alpha\in\mathcal{P}}\frac{2z}{z^2-\alpha^2}. \tag*{(98)} L ( 1/2 + z ) = B ( z ) − z m 0 − α ∈ P ∑ z 2 − α 2 2 z . (98) Here m 0 m_0 m 0 P \mathcal{P} P
B ( z ) = { 1 2 log ( ∣ D ∣ / π ) + 1 2 ψ ( ( 1 / 2 + a D + z ) / 2 ) , χ D , 1 2 log Q d + ψ ( 1 / 2 + z ) − log ( 2 π ) , K , ψ ( k / 2 + z ) − log ( 2 π ) , H k . B(z)=
\begin{cases}
\frac{1}{2}\log(|D|/\pi)+\frac{1}{2}\psi((1/2+a_D+z)/2), & \chi_D,\\
\frac{1}{2}\log Q_d+\psi(1/2+z)-\log(2\pi), & K,\\
\psi(k/2+z)-\log(2\pi), & H_k.
\end{cases} B ( z ) = ⎩ ⎨ ⎧ 2 1 log ( ∣ D ∣/ π ) + 2 1 ψ (( 1/2 + a D + z ) /2 ) , 2 1 log Q d + ψ ( 1/2 + z ) − log ( 2 π ) , ψ ( k /2 + z ) − log ( 2 π ) , χ D , K , H k . In particular B = O ( L ) B=O(L) B = O ( L ) z = 0 z=0 z = 0 (97) . For root number − 1 -1 − 1 z m 0 z^{m_0} z m 0 m 0 m_0 m 0 m 0 / z m_0/z m 0 / z m 0 = 1 m_0=1 m 0 = 1
The contour formula is
L ( s ) = P ℓ ( s ) + ∑ ρ K ℓ ( ρ − s ) + E t r ( s ) , (99) \mathcal{L}(s)=P_\ell(s)+\sum_{\rho}K_\ell(\rho-s)+E_{\rm tr}(s), \tag*{(99)} L ( s ) = P ℓ ( s ) + ρ ∑ K ℓ ( ρ − s ) + E tr ( s ) , (99) with starred versions for K K K ℜ z = 2 \Re z=2 ℜ z = 2 ≫ 1 / log ( Q ( 2 + T ) ) \gg1/\log(Q(2+T)) ≫ 1/ log ( Q ( 2 + T )) (97) and the order-one product bound the logarithmic derivative there by a fixed power of the logarithm. The kernels have vertical decay ∣ z ∣ − 3 |z|^{-3} ∣ z ∣ − 3 ∣ z ∣ − 3 − k ℓ |z|^{-3-k\ell} ∣ z ∣ − 3 − k ℓ − L ′ / L -\mathcal{L}'/\mathcal{L} − L ′ / L (99) is positive when the argument is ρ − s \rho-s ρ − s ( s − ρ ) 3 (s-\rho)^3 ( s − ρ ) 3 O ( ℓ − 2 e − ℓ ℜ s ) O(\ell^{-2}e^{-\ell\Re s}) O ( ℓ − 2 e − ℓ ℜ s ) O ( ℓ − 2 e − ℓ ℜ s / 2 ) O(\ell^{-2}e^{-\ell\Re s/2}) O ( ℓ − 2 e − ℓ ℜ s /2 ) 0 , − 1 , − 2 , … 0,-1,-2,\ldots 0 , − 1 , − 2 , …
Lemma 5.7. For the rational and weight families, outside the detector in Lemma 5.5 or (92) , the function L L L s = 1 / 2 + δ + i t s=1/2+\delta+it s = 1/2 + δ + i t ∣ t ∣ ≤ T ∗ |t|\leq T_* ∣ t ∣ ≤ T ∗ 4 / ℓ ≤ δ ≤ 3 / 2 4/\ell\leq\delta\leq3/2 4/ ℓ ≤ δ ≤ 3/2
L ( s ) = P ℓ ( s ) + O ( e − δ ℓ { L + ∣ P ℓ ( 1 / 2 + 4 / ℓ + i t ) ∣ } ) . (100) \mathcal{L}(s)=P_\ell(s)+O\left(e^{-\delta\ell}\{L+|P_\ell(1/2+4/\ell+it)|\}\right). \tag*{(100)} L ( s ) = P ℓ ( s ) + O ( e − δ ℓ { L + ∣ P ℓ ( 1/2 + 4/ ℓ + i t ) ∣ } ) . (100) For K K K
C ( T ∗ L + r ) ( 1 + T ∗ 6 L 5 ) e − c r + C X − c 0 , r = L / ℓ → ∞ , T ∗ ≤ τ / 4 , (101) C(T_*L+r)(1+T_*^6L^5)e^{-cr}+CX^{-c_0},\qquad r=L/\ell\to\infty,\qquad T_*\leq\tau/4, \tag*{(101)} C ( T ∗ L + r ) ( 1 + T ∗ 6 L 5 ) e − cr + C X − c 0 , r = L / ℓ → ∞ , T ∗ ≤ τ /4 , (101) the analogous assertion holds for 8 / ℓ ≤ δ ≤ 3 / 2 8/\ell\leq\delta\leq3/2 8/ ℓ ≤ δ ≤ 3/2 P ℓ ∗ P_\ell^* P ℓ ∗ e − δ ℓ / 2 e^{-\delta\ell/2} e − δ ℓ /2 P ℓ P_\ell P ℓ e − δ ℓ e^{-\delta\ell} e − δ ℓ T ∗ ≤ L − 5 / 6 T_*\leq L^{-5/6} T ∗ ≤ L − 5/6 1 + T ∗ 6 L 5 1+T_*^6L^5 1 + T ∗ 6 L 5 (101) is bounded.
Proof. At the anchor s 0 = 1 / 2 + a + i t s_0=1/2+a+it s 0 = 1/2 + a + i t S 0 = ∑ ρ a / ∣ s 0 − ρ ∣ 2 S_0=\sum_\rho a/|s_0-\rho|^2 S 0 = ∑ ρ a /∣ s 0 − ρ ∣ 2 T 0 = ∑ ρ ( a − β ) / ∣ s 0 − ρ ∣ 2 T_0=\sum_\rho(a-\beta)/|s_0-\rho|^2 T 0 = ∑ ρ ( a − β ) /∣ s 0 − ρ ∣ 2 T 0 T_0 T 0 S 0 S_0 S 0
a 2 + ( γ − t ) 2 − β 2 a 2 + ( γ − t ) 2 + β 2 ≥ 3 / 5 \frac{a^2+(\gamma-t)^2-\beta^2}{a^2+(\gamma-t)^2+\beta^2}\geq3/5 a 2 + ( γ − t ) 2 + β 2 a 2 + ( γ − t ) 2 − β 2 ≥ 3/5 whenever ∣ β ∣ ≤ a / 2 |\beta|\leq a/2 ∣ β ∣ ≤ a /2 ∣ γ − t ∣ > 2 ∣ β ∣ |\gamma-t|>2|\beta| ∣ γ − t ∣ > 2∣ β ∣ a = 4 / ℓ a=4/\ell a = 4/ ℓ A = ℓ a = 4 A=\ell a=4 A = ℓ a = 4 B = ℓ β B=\ell\beta B = ℓ β C = ℓ ( γ − t ) C=\ell(\gamma-t) C = ℓ ( γ − t ) K ℓ K_\ell K ℓ B ≤ 2 B\leq2 B ≤ 2
2 A 2 ( e − A / 2 + 2 e − A + e − 3 A / 2 ) < 1 / 32. \frac{2}{A^2}\left(e^{-A/2}+2e^{-A}+e^{-3A/2}\right)<1/32. A 2 2 ( e − A /2 + 2 e − A + e − 3 A /2 ) < 1/32. For B > 2 B>2 B > 2 ∣ C ∣ > e 3 B |C|>e^{3B} ∣ C ∣ > e 3 B ( e − 2 A + 2 e − 5 A / 2 + e − 3 A ) / A < 1 / 32 (e^{-2A}+2e^{-5A/2}+e^{-3A})/A<1/32 ( e − 2 A + 2 e − 5 A /2 + e − 3 A ) / A < 1/32 (98) and (99) therefore absorb the zero sum with coefficient 3 / 5 − 1 / 32 = 91 / 160 3/5-1/32=91/160 3/5 − 1/32 = 91/160 S 0 ≪ L + ∣ P ℓ ( s 0 ) ∣ S_0\ll L+|P_\ell(s_0)| S 0 ≪ L + ∣ P ℓ ( s 0 ) ∣ A 1 = ℓ δ > 4 A_1=\ell\delta>4 A 1 = ℓ δ > 4 e − A 1 e^{-A_1} e − A 1 B ≤ 2 B\leq2 B ≤ 2 ∣ s − ρ ∣ ≥ ∣ s 0 − ρ ∣ |s-\rho|\geq|s_0-\rho| ∣ s − ρ ∣ ≥ ∣ s 0 − ρ ∣ ℓ ∣ s − ρ ∣ ≥ A 1 / 2 \ell|s-\rho|\geq A_1/2 ℓ ∣ s − ρ ∣ ≥ A 1 /2 B > 2 B>2 B > 2 ∣ C ∣ > e 3 B |C|>e^{3B} ∣ C ∣ > e 3 B (100) . The inequalities hold up to δ = 3 / 2 \delta= 3/2 δ = 3/2
For the starred kernel exclude the following two sets of zeros:
4 / ℓ < β ≤ β 0 , ∣ γ ∣ ≤ T ∗ + min ( τ , ℓ − 1 e 4 ℓ β ) , β > β 0 , ∣ γ ∣ ≤ Y = X θ β 0 . \begin{aligned}
4/\ell&< \beta\le\beta_0,\quad|\gamma| \le T_*+\min(\tau,\ell^{-1}e^{4\ell\beta}),\\
\beta&> \beta_0,\quad|\gamma| \le Y=X^{\theta\beta_0}.
\end{aligned} 4/ ℓ β < β ≤ β 0 , ∣ γ ∣ ≤ T ∗ + min ( τ , ℓ − 1 e 4 ℓ β ) , > β 0 , ∣ γ ∣ ≤ Y = X θ β 0 . The second has probability O ( X − c 0 ) O(X^{-c_0}) O ( X − c 0 ) (89) . Apply (88) to the first in bins j / ℓ < β ≤ ( j + 1 ) / ℓ j/\ell< \beta\le(j+1)/\ell j / ℓ < β ≤ ( j + 1 ) / ℓ j ≥ 4 j \ge4 j ≥ 4 2 ( T ∗ + min ( τ , ℓ − 1 e 4 ( j + 1 ) ) ) 2(T_*+\min(\tau,\ell^{-1}e^{4(j+1)})) 2 ( T ∗ + min ( τ , ℓ − 1 e 4 ( j + 1 ) )) C ( T ∗ L + r ) e 4 ( j + 1 ) C(T_*L+r)e^{4(j+1)} C ( T ∗ L + r ) e 4 ( j + 1 ) C ( 1 + T ∗ 6 L 5 ) ( 1 + r 6 / L ) e 24 ( j + 1 ) C(1+T_*^6L^5)(1+r^6/L)e^{24(j+1)} C ( 1 + T ∗ 6 L 5 ) ( 1 + r 6 / L ) e 24 ( j + 1 ) r → ∞ r \to\infty r → ∞ 1 + r 6 / L 1+r^6/L 1 + r 6 / L (101) .
Now take a = 8 / ℓ a=8/\ell a = 8/ ℓ 3 / 5 3/5 3/5 β ≤ β 0 \beta\le\beta_0 β ≤ β 0 ∣ γ − t ∣ > τ |\gamma-t|>\tau ∣ γ − t ∣ > τ (96) and (97) , O ( L e − ℓ δ / 2 − τ ℓ / 20 ) O(Le^{-\ell\delta/2-\tau\ell/20}) O ( L e − ℓ δ /2 − τ ℓ /20 ) β > β 0 \beta>\beta_0 β > β 0 ∣ γ − t ∣ > Y / 2 |\gamma-t|>Y/2 ∣ γ − t ∣ > Y /2
O ( ℓ − 2 e 7 ℓ / 4 − ℓ δ / 2 Y − 2 log ( Q Y ) ) ≪ L e − ℓ δ / 2 X − c , O\left(\ell^{-2}e^{7\ell/4-\ell\delta/2}Y^{-2}\log(QY)\right)\ll Le^{-\ell\delta/2}X^{-c}, O ( ℓ − 2 e 7 ℓ /4 − ℓ δ /2 Y − 2 log ( Q Y ) ) ≪ L e − ℓ δ /2 X − c , because ℓ = o ( L ) \ell=o(L) ℓ = o ( L ) A = 8 A=8 A = 8 (95) . If B ≤ 4 B\le4 B ≤ 4 8 A − 2 e − A / 4 = e − 2 / 8 < 1 / 32 8A^{-2}e^{-A/4}=e^{-2}/8<1/32 8 A − 2 e − A /4 = e − 2 /8 < 1/32 B > 4 B>4 B > 4 ∣ C ∣ > e 4 B |C|>e^{4B} ∣ C ∣ > e 4 B 4 A − 1 e − 9 A / 4 < 1 / 32 4A^{-1}e^{-9A/4}<1/32 4 A − 1 e − 9 A /4 < 1/32 A 1 = ℓ δ ≥ 8 A_1=\ell\delta\ge8 A 1 = ℓ δ ≥ 8 C e − A 1 / 2 Ce^{-A_1/2} C e − A 1 /2 ∣ s − ρ ∣ ≥ ∣ s 0 − ρ ∣ |s-\rho|\ge|s_0-\rho| ∣ s − ρ ∣ ≥ ∣ s 0 − ρ ∣ ∣ C ∣ > e 4 B |C|>e^{4B} ∣ C ∣ > e 4 B
Model bounds and the central interval
In all three families set
G D ( w ) = e − w L D ( 1 / 2 + e − w ) , G E ( w ) = e − w ∑ p , j ≥ 1 a p , j ( log p ) p − j ( 1 / 2 + e − w ) . G_D(w)=e^{-w}L_D(1/2+e^{-w}),\qquad G_E(w)=e^{-w}\sum_{p,j\ge1}a_{p,j}(\log p)p^{-j(1/2+e^{-w})}. G D ( w ) = e − w L D ( 1/2 + e − w ) , G E ( w ) = e − w p , j ≥ 1 ∑ a p , j ( log p ) p − j ( 1/2 + e − w ) . with prime ideals and their norms in the Hecke case. In the quadratic cases a p , j = X p j a_{p,j}=X_p^j a p , j = X p j a p , j = A j ( X p ) a_{p,j}=A_j(X_p) a p , j = A j ( X p )
E G E ( t ) = μ + O ( e − t ) , Cov ( G E ( t ) , G E ( s ) ) = 1 4 sech 2 ( ( t − s ) / 2 ) + O ( e − t − s ) , μ = { 1 / 2 , quadratic cases , − 1 / 2 , H k . (102) \begin{aligned}
\mathbb{E}G_E(t)&=\mu+O(e^{-t}),\\
\operatorname{Cov}(G_E(t),G_E(s))&=\frac{1}{4}\operatorname{sech}^{2}((t-s)/2)+O(e^{-t-s}),\\
\mu&=\begin{cases}
1/2,&\text{quadratic cases},\\
-1/2,&H_k.
\end{cases} \tag*{(102)}
\end{aligned} E G E ( t ) Cov ( G E ( t ) , G E ( s )) μ = μ + O ( e − t ) , = 4 1 sech 2 (( t − s ) /2 ) + O ( e − t − s ) , = { 1/2 , − 1/2 , quadratic cases , H k . (102) We indicate the checks that distinguish these applications. The prime sum is an analytic L 2 L^2 L 2 K K K ∑ ( log p ) p − 1 − z = z − 1 + O ( 1 ) \sum(\log p)p^{-1-z}=z^{-1}+O(1) ∑ ( log p ) p − 1 − z = z − 1 + O ( 1 ) ∑ ( log p ) 2 p − 1 − z = z − 2 + O ( 1 ) \sum(\log p)^2p^{-1-z}=z^{-2}+O(1) ∑ ( log p ) 2 p − 1 − z = z − 2 + O ( 1 ) 1 + O ( p − 1 ) 1+O(p^{-1}) 1 + O ( p − 1 ) − 1 + O ( p − 1 ) -1+O(p^{-1}) − 1 + O ( p − 1 ) (66) ; these give the two means in (102) . The first powers give the covariance. All remaining covariance terms are absolutely summable. Whole-prime blocks have range at most C ∣ a ∣ ( log p ) p − 1 / 2 − ℜ a / ( 1 − p − 1 / 2 − ℜ a ) C|a|(\log p)p^{-1/2-\Re a}/(1-p^{-1/2-\Re a}) C ∣ a ∣ ( log p ) p − 1/2 − ℜ a / ( 1 − p − 1/2 − ℜ a ) O ( 1 ) O(1) O ( 1 ) O ( ∣ a ∣ 3 ) O(|a|^3) O ( ∣ a ∣ 3 ) (65) .
In particular, for a fixed finite number of points and every q ≥ 2 q\ge2 q ≥ 2 L q L^q L q O ( q ) O(\sqrt q) O ( q ) q q q a 0 = 4 / ℓ a_0=4/\ell a 0 = 4/ ℓ 8 / ℓ 8/\ell 8/ ℓ
∥ ℓ − 1 P ℓ ( 1 / 2 + a 0 + i t ) ∥ q ≪ q 1 , (103) \left\|\ell^{-1}P_\ell(1/2+a_0+it)\right\|_q\ll_q 1, \tag*{(103)} ℓ − 1 P ℓ ( 1/2 + a 0 + i t ) q ≪ q 1 , (103) uniformly in the height t t t O q ( ℓ ) O_q(\ell) O q ( ℓ ) 2 m 2m 2 m X − κ exp ( C m ℓ + O ( m log m ) ) X^{-\kappa}\exp(Cm\ell+O(m\log m)) X − κ exp ( C m ℓ + O ( m log m )) X X X k k k e i v Y e^{ivY} e i v Y 2 m − 1 2m-1 2 m − 1
2 ( C ∣ v ∣ / m ) 2 m + X − κ exp ( C m ( ℓ + log ( 2 + ∣ v ∣ ) + log m ) ) . (104) 2(C|v|/\sqrt{m})^{2m}+X^{-\kappa}\exp(Cm(\ell+\log(2+|v|)+\log m)). \tag*{(104)} 2 ( C ∣ v ∣/ m ) 2 m + X − κ exp ( C m ( ℓ + log ( 2 + ∣ v ∣ ) + log m )) . (104) The actual even remainder is transferred before it is bounded; this is why the estimate is valid on the original family. This is the argument of Proposition 2.16 , with the actual arithmetic norms verified here.
The ternary models have a uniform negative one-point moment, by Lemma 2.7 and Corollary 2.8 :
E ∣ G E ( t ) ∣ − a 0 ≤ C ( t ≥ log 2 ) (105) \mathbb{E}|G_E(t)|^{-a_0}\le C \qquad(t\ge\log2) \tag*{(105)} E ∣ G E ( t ) ∣ − a 0 ≤ C ( t ≥ log 2 ) (105) for some a 0 > 0 a_0>0 a 0 > 0 3 / 4 3/4 3/4 ( 1 + i ) (1+i) ( 1 + i )
P ( ∣ G E ( t ) − v ∣ ≤ h ) ≪ e t h , P ( ∣ G E ( t ) − v ∣ ≤ h ) ≪ h + e − t / 4 . (106) \mathbb{P}(|G_E(t)-v|\le h)\ll e^t h,\qquad
\mathbb{P}(|G_E(t)-v|\le h)\ll h+e^{-t/4}. \tag*{(106)} P ( ∣ G E ( t ) − v ∣ ≤ h ) ≪ e t h , P ( ∣ G E ( t ) − v ∣ ≤ h ) ≪ h + e − t /4 . (106) For the first inequality condition on all primes except 2 2 2 a = e − t a=e^{-t} a = e − t r = 2 − 1 / 2 − a r=2^{-1/2-a} r = 2 − 1/2 − a a log 2 ( r x − 2 r 2 ) / ( 1 − r x + r 2 ) a\log2\,(rx-2r^2)/(1-rx+r^2) a log 2 ( r x − 2 r 2 ) / ( 1 − r x + r 2 ) a a a log 2 r ( 1 − r 2 ) / ( 1 − r x + r 2 ) 2 ≫ 1 \log2\,r(1-r^2)/(1-rx+r^2)^2\gg1 log 2 r ( 1 − r 2 ) / ( 1 − r x + r 2 ) 2 ≫ 1 [ − 2 , 2 ] [-2,2] [ − 2 , 2 ] (65) is bounded, giving density O ( a − 1 ) O(a^{-1}) O ( a − 1 ) e − t / 4 e^{-t/4} e − t /4 e − t e^{-t} e − t e − 2 t e^{-2t} e − 2 t e − 3 t e^{-3t} e − 3 t
Take now ν = A H \nu=A\sqrt{H} ν = A H Δ = ν / L \Delta=\nu/L Δ = ν / L A > 0 A>0 A > 0 ℓ = L / ν \ell=L/\sqrt{\nu} ℓ = L / ν c Δ ≤ δ ≤ C Δ c\Delta\le\delta\le C\Delta c Δ ≤ δ ≤ C Δ Lemma 5.7 imply
∥ 1 g o o d δ ( L − P ℓ ) ( 1 / 2 + δ ) ∥ 1 ≪ ν e − c ν = o ( 1 ) . \left\|\mathbf{1}_{\mathrm{good}}\delta(\mathcal{L}-P_\ell)(1/2+\delta)\right\|_1\ll\nu e^{-c\sqrt{\nu}}=o(1). ∥ 1 good δ ( L − P ℓ ) ( 1/2 + δ ) ∥ 1 ≪ ν e − c ν = o ( 1 ) . Here (103) was used with its normalization. The detector probabilities tend to zero, including the factor H 20 H^{20} H 20 (104) choose m = ⌊ ν 1 / 8 ⌋ m=\lfloor\nu^{1/8}\rfloor m = ⌊ ν 1/8 ⌋ ∣ v ∣ ≤ ν 1 / 64 |v|\le\nu^{1/64} ∣ v ∣ ≤ ν 1/64 ∣ v ∣ / m ≪ ν − 3 / 64 |v|/\sqrt{m}\ll\nu^{-3/64} ∣ v ∣/ m ≪ ν − 3/64 m ℓ / L ≪ ν − 3 / 8 m\ell/L\ll\nu^{-3/8} m ℓ / L ≪ ν − 3/8
δ L D ( 1 / 2 + δ ) ⟹ N ( μ , 1 / 4 ) . (107) \delta\mathcal{L}_D(1/2+\delta)\Longrightarrow N(\mu,1/4). \tag*{(107)} δ L D ( 1/2 + δ ) ⟹ N ( μ , 1/4 ) . (107) For U U U [ 1 , 2 ] [1,2] [ 1 , 2 ]
P ( ∣ Δ L D ( 1 / 2 + U Δ ) ∣ < e − H ) = o ( 1 ) . (108) \mathbb{P}(|\Delta\mathcal{L}_D(1/2+U\Delta)|<e^{-\sqrt{H}})=o(1). \tag*{(108)} P ( ∣Δ L D ( 1/2 + U Δ ) ∣ < e − H ) = o ( 1 ) . (108) To justify the shrinking threshold, first bound it by the probability of a fixed interval of radius ε \varepsilon ε (107) , and then let ε \varepsilon ε U ∈ [ 1 , 2 ] U\in[1,2] U ∈ [ 1 , 2 ]
Proposition 5.8. In each of the three families,
R L ′ D m u l t ( [ 1 / 2 , 1 / 2 + Δ ] ) = O P ( H ) . R^{\mathrm{mult}}_{L'D}([1/2,1/2+\Delta])=O_{\mathbb{P}}(H). R L ′ D mult ([ 1/2 , 1/2 + Δ ]) = O P ( H ) . Proof. Remove the wedge exceptional sets in (82) , (90) , or (93) , and, in the rational case, the O ( X − 1 / 2 ) O(X^{-1/2}) O ( X − 1/2 ) ∣ D ∣ ≤ X |D|\le\sqrt{X} ∣ D ∣ ≤ X ∣ β ∣ ≤ Δ / 2 |\beta|\le\Delta/2 ∣ β ∣ ≤ Δ/2 ∣ γ ∣ > 2 ∣ β ∣ |\gamma|>2|\beta| ∣ γ ∣ > 2∣ β ∣ Δ \Delta Δ
6 5 Δ Δ 2 + β 2 + γ 2 . \frac{6}{5}\frac{\Delta}{\Delta^2+\beta^2+\gamma^2}. 5 6 Δ 2 + β 2 + γ 2 Δ . This follows by writing D 0 = Δ 2 + β 2 + γ 2 D_0 = \Delta^2 + \beta^2 + \gamma^2 D 0 = Δ 2 + β 2 + γ 2 3 D 0 / 5 3D_0/5 3 D 0 /5 D 0 2 D_0^2 D 0 2
K D = ∑ ρ Δ 2 Δ 2 + ∣ ρ − 1 / 2 ∣ 2 ≤ 5 3 Δ ℜ { B ( Δ ) − L D ( 1 / 2 + Δ ) } K_D = \sum_{\rho} \frac{\Delta^2}{\Delta^2 + \lvert\rho- 1/2\rvert^2} \le\frac{5}{3}\Delta\Re\{B(\Delta)-L_D(1/2+\Delta)\} K D = ρ ∑ Δ 2 + ∣ ρ − 1/2 ∣ 2 Δ 2 ≤ 3 5 Δℜ { B ( Δ ) − L D ( 1/2 + Δ )} has restricted expectation O ( ν + 1 ) O(\nu+1) O ( ν + 1 ) B = O ( L ) B=O(L) B = O ( L ) N D N_D N D ∣ ρ − 1 / 2 ∣ ≤ 12 Δ \lvert\rho-1/2\rvert\le12\Delta ∣ ρ − 1/2 ∣ ≤ 12Δ W D = Δ 2 ∑ ∣ α ∣ > 12 Δ ∣ α ∣ − 2 W_D=\Delta^2\sum_{\lvert\alpha\rvert>12\Delta}\lvert\alpha\rvert^{-2} W D = Δ 2 ∑ ∣ α ∣ > 12Δ ∣ α ∣ − 2 (98) , then N D ≤ 145 K D N_D\le145K_D N D ≤ 145 K D W D ≤ 145 K D / 288 W_D\le145K_D/288 W D ≤ 145 K D /288 (108) , that lemma bounds the probability of a count exceeding η H \eta H η H
o ( 1 ) + C { ν + 1 + log ( ν + 2 ) + H } η H = o ( 1 ) . o(1)+\frac{C\{\nu+1+\log(\nu+2)+\sqrt{H}\}}{\eta H}=o(1). o ( 1 ) + η H C { ν + 1 + log ( ν + 2 ) + H } = o ( 1 ) . The central pole m 0 / z m_0/z m 0 / z K D K_D K D N D N_D N D m 0 m_0 m 0 m 0 − 1 m_0-1 m 0 − 1 − 1 -1 − 1 L ′ ≢ 0 L'\not\equiv0 L ′ ≡ 0 λ f ( 2 ) = 0 \lambda_f(2)=0 λ f ( 2 ) = 0 λ f ( 4 ) = − 1 \lambda_f(4)=-1 λ f ( 4 ) = − 1
The short bridge
We will need a growing-dimensional small-ball test for the weight family as well as for the ternary models. The following variant makes the necessary hypothesis on continuous blocks explicit.
Lemma 5.9. Let Y = μ + ∑ ξ p ∈ R r Y=\mu+\sum\xi_p\in\mathbb{R}^r Y = μ + ∑ ξ p ∈ R r c I ≤ Σ ≤ C I cI\le\Sigma\le CI c I ≤ Σ ≤ C I ∥ μ ∥ 2 ≤ C r \lVert\mu\rVert^2\le Cr ∥ μ ∥ 2 ≤ C r ∥ ξ p ∥ ≤ β p ≤ 1 \lVert\xi_p\rVert\le\beta_p\le1 ∥ ξ p ∥ ≤ β p ≤ 1 h ∈ R r h\in\mathbb{R}^r h ∈ R r
∑ p ( e s s s u p h ⋅ ξ p − e s s i n f h ⋅ ξ p ) 2 ≤ C ∥ h ∥ 2 . \sum_p\left(\mathop{\mathrm{ess\,sup}} h\cdot\xi_p-\mathop{\mathrm{ess\,inf}} h\cdot\xi_p\right)^2\le C\lVert h\rVert^2. p ∑ ( ess sup h ⋅ ξ p − ess inf h ⋅ ξ p ) 2 ≤ C ∥ h ∥ 2 . For all sufficiently large r r r ϵ 0 > 0 \epsilon_0>0 ϵ 0 > 0 A 0 = tr Σ + ∥ μ ∥ 2 A_0=\operatorname{tr}\Sigma+\lVert\mu\rVert^2 A 0 = tr Σ + ∥ μ ∥ 2 m = ⌊ ϵ 0 r ⌋ m=\lfloor\epsilon_0r\rfloor m = ⌊ ϵ 0 r ⌋ Q ( y ) = ( ( A 0 − ∥ y ∥ 2 ) / ( A 0 − 1 ) ) 2 m Q(y)=((A_0-\lVert y\rVert^2)/(A_0-1))^{2m} Q ( y ) = (( A 0 − ∥ y ∥ 2 ) / ( A 0 − 1 ) ) 2 m
1 { ∥ y ∥ ≤ 1 } ≤ Q ( y ) , E Q ( Y ) ≤ e − c r + e C r ∑ p β p 3 . \mathbf{1}_{\{\lVert y\rVert\le1\}}\le Q(y),\qquad\mathbb{E}Q(Y)\le e^{-cr}+e^{Cr}\sum_p\beta_p^3. 1 {∥ y ∥ ≤ 1 } ≤ Q ( y ) , E Q ( Y ) ≤ e − cr + e C r p ∑ β p 3 . Proof. For sufficiently large r r r A 0 − 1 ≥ c r / 2 A_0-1\ge cr/2 A 0 − 1 ≥ cr /2 m ≥ 1 m\ge1 m ≥ 1 L q L^q L q O ( r q + q ) O(\sqrt{rq}+q) O ( r q + q ) 2 − 2 m 2^{-2m} 2 − 2 m ϵ 0 \epsilon_0 ϵ 0 E e h ⋅ W ≤ e C ∥ h ∥ 2 \mathbb{E}e^{h\cdot W}\le e^{C\lVert h\rVert^2} E e h ⋅ W ≤ e C ∥ h ∥ 2 1 / 2 1/2 1/2 5 r 5^r 5 r ∥ W ∥ L q ≪ r + q \lVert W\rVert_{L^q}\ll\sqrt{r}+\sqrt{q} ∥ W ∥ L q ≪ r + q p = 2 m p=2m p = 2 m q ( y ) = A 0 − ∥ y ∥ 2 q(y)=A_0-\lVert y\rVert^2 q ( y ) = A 0 − ∥ y ∥ 2
( A 0 − 1 ) − p { p ( p − 1 ) ( p − 2 ) q ( y ) p − 3 ( − 2 y ⋅ h ) 3 + 3 p ( p − 1 ) q ( y ) p − 2 ( − 2 y ⋅ h ) ( − 2 ∥ h ∥ 2 ) } . (A_0-1)^{-p}\{p(p-1)(p-2)q(y)^{p-3}(-2y\cdot h)^3+3p(p-1)q(y)^{p-2}(-2y\cdot h)(-2\lVert h\rVert^2)\}. ( A 0 − 1 ) − p { p ( p − 1 ) ( p − 2 ) q ( y ) p − 3 ( − 2 y ⋅ h ) 3 + 3 p ( p − 1 ) q ( y ) p − 2 ( − 2 y ⋅ h ) ( − 2 ∥ h ∥ 2 )} . Its absolute value is bounded by e C r ∥ h ∥ 3 ( 1 + ∥ y ∥ / r ) 2 p + 3 e^{Cr}\lVert h\rVert^3(1+\lVert y\rVert/\sqrt{r})^{2p+3} e C r ∥ h ∥ 3 ( 1 + ∥ y ∥ / r ) 2 p + 3 ∥ g p ∥ L q ≪ q β p \lVert g_p\rVert_{L^q}\ll\sqrt{q}\,\beta_p ∥ g p ∥ L q ≪ q β p e C r β p 3 e^{Cr}\beta_p^3 e C r β p 3 r − j / 2 ( j + 3 ) ( j + 3 ) / 2 ≤ C j ( C r ) 3 / 2 r^{-j/2}(j+3)^{(j+3)/2}\le C^j(Cr)^{3/2} r − j /2 ( j + 3 ) ( j + 3 ) /2 ≤ C j ( C r ) 3/2 j = O ( r ) j=O(r) j = O ( r ) e C r e^{Cr} e C r Q Q Q A 0 − ∥ y ∥ 2 ≥ A 0 − 1 > 0 A_0-\lVert y\rVert^2\ge A_0-1>0 A 0 − ∥ y ∥ 2 ≥ A 0 − 1 > 0
Put U = H 1 / 4 U=H^{1/4} U = H 1/4
u = { ⌊ H ⌋ , F ( X ) , ⌊ H 1 / 8 ⌋ , D X , H k , d = H 8 e 2 u L , Δ = A U 2 L . (109) u=\begin{cases}
\lfloor\sqrt{H}\rfloor, & \mathcal{F}(X),\\
\lfloor H^{1/8}\rfloor, & \mathcal{D}_X,H_k,
\end{cases}
\qquad d=\frac{H^{8}e^{2u}}{L},\qquad\Delta=\frac{AU^{2}}{L}. \tag*{(109)} u = { ⌊ H ⌋ , ⌊ H 1/8 ⌋ , F ( X ) , D X , H k , d = L H 8 e 2 u , Δ = L A U 2 . (109) Here A A A [ 1 / 2 + Δ , 1 / 2 + e 5 d ] [1/2+\Delta,1/2+e^{5}d] [ 1/2 + Δ , 1/2 + e 5 d ]
Proposition 5.10. For this choice of A A A o P ( H ) o_{\mathrm{P}}(H) o P ( H )
Proof. Cover its logarithmic image by half-step cells, with centres t j t_j t j δ j = e − t j \delta_j=e^{-t_j} δ j = e − t j 1 / 2 1/2 1/2 n = O ( u + log H ) n=O(u+\log H) n = O ( u + log H ) S j = δ j L S_j=\delta_jL S j = δ j L
e − 1 / 2 A U 2 ≤ S j ≤ e 11 / 2 H 8 e 2 u . (110) e^{-1/2}AU^{2}\leq S_j\leq e^{11/2}H^{8}e^{2u}. \tag*{(110)} e − 1/2 A U 2 ≤ S j ≤ e 11/2 H 8 e 2 u . (110) The required heights are T ∗ = 3 δ j T_* =3\delta_j T ∗ = 3 δ j o ( L − 5 / 6 ) o(L^{-5/6}) o ( L − 5/6 ) δ j L 5 / 6 ≤ exp ( − H / 6 + O ( u + log H ) ) \delta_jL^{5/6}\leq\exp(-H/6+O(u+\log H)) δ j L 5/6 ≤ exp ( − H /6 + O ( u + log H ))
First use ℓ j = L / S j \ell_j=L/\sqrt{S_j} ℓ j = L / S j
C H 20 ( 1 + 3 S j ) e − c S j + C e − 4 H , C ( S j + S j ) e − c S j + C X − c 0 , C ( S j + S j ) e − c S j . CH^{20}(1+3\sqrt{S_j})e^{-c\sqrt{S_j}}+Ce^{-4H},\qquad C(S_j+\sqrt{S_j})e^{-c\sqrt{S_j}}+CX^{-c_0},\qquad C(S_j+\sqrt{S_j})e^{-c\sqrt{S_j}}. C H 20 ( 1 + 3 S j ) e − c S j + C e − 4 H , C ( S j + S j ) e − c S j + C X − c 0 , C ( S j + S j ) e − c S j . Their sum over the cells is o ( 1 ) o(1) o ( 1 ) L q L^q L q C q S j e − c S j C_qS_je^{-c\sqrt{S_j}} C q S j e − c S j (103) . Cauchy’s formula on fixed disks and the polynomial moment bounds therefore show that the sum of the positive logarithms of the large disk maxima has expectation O ( n ) = o ( H ) O(n)=o(H) O ( n ) = o ( H )
The small disk maximum requires a longer polynomial. Choose r = ⌊ ϵ U ⌋ r=\lfloor\epsilon U\rfloor r = ⌊ ϵ U ⌋ ϵ log 64 < 1 / 4 \epsilon\log64<1/4 ϵ log 64 < 1/4 v j = r + log ( 2 + S j ) v_j=r+\log(2+S_j) v j = r + log ( 2 + S j )
ℓ j ′ = C y v j δ j . (111) \ell'_j=\frac{C_yv_j}{\delta_j}. \tag*{(111)} ℓ j ′ = δ j C y v j . (111) The ratio L / ℓ j ′ = S j / ( C y v j ) L/\ell'_j=S_j/(C_yv_j) L / ℓ j ′ = S j / ( C y v j ) ≫ U \gg U ≫ U A A A
sup j log H + log S j S j / ( C y v j ) ⟶ 0. \sup_j\frac{\log H+\log S_j}{S_j/(C_yv_j)}\longrightarrow0. j sup S j / ( C y v j ) log H + log S j ⟶ 0. This follows from the monotonicity of S / ( r + log ( 2 + S ) ) S/(r+\log(2+S)) S / ( r + log ( 2 + S )) (110) . It is needed in the rational case, where log S j \log S_j log S j o ( r ) o(r) o ( r ) (111) , including the H 20 H^{20} H 20 o ( 1 ) o(1) o ( 1 )
Use the Cayley map a ( z ) = δ j ( 1 + z ) / ( 1 − z ) a(z)=\delta_j(1+z)/(1-z) a ( z ) = δ j ( 1 + z ) / ( 1 − z )
H D ( z ) = δ j ( 1 − z ) 2 L D ( 1 / 2 + a ( z ) ) , Z k = 2 k + 1 [ z k ] H D ( z ) , 0 ≤ k < r . \mathcal{H}_D(z)=\frac{\delta_j}{(1-z)^2}\mathcal{L}_D(1/2+a(z)),\qquad Z_k=\frac{2}{\sqrt{k+1}}[z^k]\mathcal{H}_D(z),\quad0\leq k<r. H D ( z ) = ( 1 − z ) 2 δ j L D ( 1/2 + a ( z )) , Z k = k + 1 2 [ z k ] H D ( z ) , 0 ≤ k < r . The map is related to logarithmic time by H D ( z ) = G D ( t j − log ( ( 1 + z ) / ( 1 − z ) ) ) / ( 1 − z 2 ) \mathcal{H}_D(z)=G_D(t_j-\log((1+z)/(1-z)))/(1-z^2) H D ( z ) = G D ( t j − log (( 1 + z ) / ( 1 − z ))) / ( 1 − z 2 ) ∣ z ∣ ≤ 1 / 64 |z|\leq1/64 ∣ z ∣ ≤ 1/64 log ( 65 / 63 ) < 1 / 8 \log(65/63)<1/8 log ( 65/63 ) < 1/8 m j m_j m j ∣ G D ∣ |G_D| ∣ G D ∣ 1 / 8 1/8 1/8 e − U e^{-U} e − U ∥ Z ∥ ≤ C r 64 r e − U < 1 / 4 \|Z\|\leq C\sqrt{r}\,64^re^{-U}<1/4 ∥ Z ∥ ≤ C r 6 4 r e − U < 1/4
Replace L D \mathcal{L}_D L D ℓ j ′ \ell'_j ℓ j ′ Y D Y_D Y D ∣ z ∣ ≤ 1 / 2 |z|\leq1/2 ∣ z ∣ ≤ 1/2 ℜ a ( z ) ≥ δ j / 3 \Re a(z)\geq\delta_j/3 ℜ a ( z ) ≥ δ j /3 ∣ ℑ a ( z ) ∣ ≤ 3 δ j |\Im a(z)|\leq3\delta_j ∣ℑ a ( z ) ∣ ≤ 3 δ j O ( v j ) O(v_j) O ( v j ) 1 / ℓ j ′ 1/\ell'_j 1/ ℓ j ′ (103) and Cauchy’s formula give
E sup ∣ t ∣ ≤ 3 δ j ∣ P ( 1 / 2 + a 0 + i t ) ∣ 2 ≪ C y v j 2 ( ℓ j ′ ) 2 . \mathbb{E}\sup_{|t|\leq3\delta_j}|P(1/2+a_0+it)|^2\ll C_yv_j^2(\ell'_j)^2. E ∣ t ∣ ≤ 3 δ j sup ∣ P ( 1/2 + a 0 + i t ) ∣ 2 ≪ C y v j 2 ( ℓ j ′ ) 2 . For fixed large b 0 b_0 b 0 δ j sup ∣ P ∣ > e b 0 v j \delta_j\sup|P|>e^{b_0v_j} δ j sup ∣ P ∣ > e b 0 v j C v j 4 e − 2 b 0 v j ≤ e − b 0 v j Cv_j^4e^{-2b_0v_j}\leq e^{-b_0v_j} C v j 4 e − 2 b 0 v j ≤ e − b 0 v j C y C_y C y b 0 b_0 b 0 C e − C y v j / 6 ( e b 0 v j + S j ) ≤ e − 30 v j Ce^{-C_yv_j/6(e^{b_0v_j}+S_j)}\leq e^{-30v_j} C e − C y v j /6 ( e b 0 v j + S j ) ≤ e − 30 v j C r 2 r C\sqrt{r}\,2^r C r 2 r ∥ Z − Y D ∥ < 1 / 4 \lVert Z-Y_D\rVert< 1/4 ∥ Z − Y D ∥ < 1/4 ∥ Y D ∥ ≤ 1 \lVert Y_D\rVert\le1 ∥ Y D ∥ ≤ 1
We verify the model-vector hypotheses instead of assuming independent Taylor coefficients. Expansion of (102) under the Cayley map gives
E Y k = 2 μ 1 2 ∣ k k + 1 + O ( δ j 2 k ) , Cov ( Y k , Y l ) = 1 k = l + O ( δ j 2 k + l ) . (112) \mathbb{E}Y_k=\frac{2\mu\,\mathbf{1}_{2\mid k}}{\sqrt{k+1}}+O(\delta_j^{2k}),\qquad\operatorname{Cov}(Y_k,Y_l)=\mathbf{1}_{k=l}+O(\delta_j^{2k+l}). \tag*{(112)} E Y k = k + 1 2 μ 1 2 ∣ k + O ( δ j 2 k ) , Cov ( Y k , Y l ) = 1 k = l + O ( δ j 2 k + l ) . (112) The squared polynomial-tail contribution is O ( r 4 e − δ j r ℓ / 6 ) O(r^4 e^{-\delta_j r^\ell/6}) O ( r 4 e − δ j r ℓ /6 ) δ j 2 r 4 r → 0 \delta_j^2r^4r\to0 δ j 2 r 4 r → 0 1 2 I ≤ Σ ≤ 2 I \frac{1}{2}I\le\Sigma\le2I 2 1 I ≤ Σ ≤ 2 I ∥ μ Y ∥ 2 ≤ C r \lVert\mu_Y\rVert^2\le Cr ∥ μ Y ∥ 2 ≤ C r ξ p = a p X p + b p ( X p 2 − θ p ) \xi_p=a_pX_p+b_p(X_p^2-\theta_p) ξ p = a p X p + b p ( X p 2 − θ p ) θ p ≥ 2 / 3 \theta_p\ge2/3 θ p ≥ 2/3 θ p ≥ 3 / 4 \theta_p\ge3/4 θ p ≥ 3/4 ∑ a p a p T ≪ I \sum a_pa_p^{\mathsf T}\ll I ∑ a p a p T ≪ I ∑ ∥ b p ∥ 2 = o ( 1 ) \sum\lVert b_p\rVert^2=o(1) ∑ ∥ b p ∥ 2 = o ( 1 ) max ∥ ξ p ∥ ≤ δ j e C r = o ( 1 ) \max\lVert\xi_p\rVert\le\delta_j e^{Cr}=o(1) max ∥ ξ p ∥ ≤ δ j e C r = o ( 1 ) ∑ ∥ ξ p ∥ 3 ≤ δ j 3 e C r \sum\lVert\xi_p\rVert^3\le\delta_j^3e^{Cr} ∑ ∥ ξ p ∥ 3 ≤ δ j 3 e C r
For the weight model write instead ξ p = a p X p + η p ( X p ) \xi_p=a_pX_p+\eta_p(X_p) ξ p = a p X p + η p ( X p ) η p \eta_p η p ∣ z ∣ = 1 / 2 |z|=1/2 ∣ z ∣ = 1/2
∥ η p ∥ ∞ ≤ b p = C δ j r 2 r ( log p ) p − 1 − 2 δ j / 3 , ∥ a p ∥ ≤ C δ j r 2 r ( log p ) p − 1 / 2 − δ j / 3 . \lVert\eta_p\rVert_\infty\le b_p=C\delta_j\sqrt{r}\,2^r(\log p)p^{-1-2\delta_j/3},\qquad\lVert a_p\rVert\le C\delta_j\sqrt{r}\,2^r(\log p)p^{-1/2-\delta_j/3}. ∥ η p ∥ ∞ ≤ b p = C δ j r 2 r ( log p ) p − 1 − 2 δ j /3 , ∥ a p ∥ ≤ C δ j r 2 r ( log p ) p − 1/2 − δ j /3 . Here ∑ b p 2 = o ( 1 ) \sum b_p^2=o(1) ∑ b p 2 = o ( 1 ) L 2 L^2 L 2
∑ p E X p 2 ( h ⋅ a p ) 2 ≤ 2 Var ( h ⋅ Y ) + 2 ∑ p E ∣ h ⋅ η p ∣ 2 ≪ ∥ h ∥ 2 . \sum_p\mathbb{E}X_p^2(h\cdot a_p)^2\le2\operatorname{Var}(h\cdot Y)+2\sum_p\mathbb{E}|h\cdot\eta_p|^2\ll\lVert h\rVert^2. p ∑ E X p 2 ( h ⋅ a p ) 2 ≤ 2 Var ( h ⋅ Y ) + 2 p ∑ E ∣ h ⋅ η p ∣ 2 ≪ ∥ h ∥ 2 . Since E X p 2 ≥ 1 \mathbb{E}X_p^2\ge1 E X p 2 ≥ 1 4 ∣ h ⋅ a p ∣ + 2 ∥ h ∥ b p 4|h\cdot a_p|+2\lVert h\rVert b_p 4∣ h ⋅ a p ∣ + 2 ∥ h ∥ b p ∑ β p 3 ≤ δ j 3 e C r \sum\beta_p^3\le\delta_j^3e^{Cr} ∑ β p 3 ≤ δ j 3 e C r o ( 1 ) o(1) o ( 1 ) X p X_p X p
It follows in every family that E Q ( Y E ) ≤ e − c r + δ j 3 e C r \mathbb{E}Q(Y_E)\le e^{-cr}+\delta_j^3e^{Cr} E Q ( Y E ) ≤ e − cr + δ j 3 e C r Q Q Q 4 m 4m 4 m 2 m 2m 2 m C 2 r exp ( C N ℓ j ′ ) C2^r\exp(C_N\ell'_j) C 2 r exp ( C N ℓ j ′ ) C N = 9 / 2 , 8 , 7 C_N=9/2,8,7 C N = 9/2 , 8 , 7
4 C N m ℓ j ′ + O ( m r + m log r + m ) . (113) 4C_Nm\ell'_j+O(mr+m\log r+m). \tag*{(113)} 4 C N m ℓ j ′ + O ( m r + m log r + m ) . (113) Choose constants in this order: the arithmetic exponent κ \kappa κ ϵ \epsilon ϵ ϵ 0 \epsilon_0 ϵ 0 b 0 b_0 b 0 C y C_y C y A A A
m ℓ j ′ L ≪ r ( r + log ( 2 + S j ) ) S j ≪ ϵ 2 A + o ( 1 ) . \frac{m\ell'_j}{L}\ll\frac{r(r+\log(2+S_j))}{S_j}\ll\frac{\epsilon^2}{A}+o(1). L m ℓ j ′ ≪ S j r ( r + log ( 2 + S j )) ≪ A ϵ 2 + o ( 1 ) . Increasing A A A (113) smaller than κ L / 4 \kappa L/4 κ L /4 o ( L ) o(L) o ( L ) X − κ / 2 X^{-\kappa/2} X − κ /2 k − κ / 2 k^{-\kappa/2} k − κ /2 C e − c U + e − H + X − κ / 2 Ce^{-cU}+e^{-H}+X^{-\kappa/2} C e − c U + e − H + X − κ /2 n n n
On the retained set all depths log + ( 1 / m j ) \log^+(1/m_j) log + ( 1/ m j ) U U U n U = o ( H ) nU=o(H) n U = o ( H ) O ( H 3 / 4 + H 1 / 4 log H ) O(H^{3/4}+H^{1/4}\log H) O ( H 3/4 + H 1/4 log H ) O ( n ) O(n) O ( n ) L L L G D G_D G D L D ′ L'_D L D ′
The bulk count with retained local moments
We provide the transfer and counting argument on the unconditional good sets just constructed. In particular an approximation under GRH is not being used for this step.
Let d , u d,u d , u (109) , and set
b H = − log d = H − 8 log H − 2 u , ℓ B = C y H d = C y L H 7 e 2 u , a H = H 1 / 4 . (114) b_H=-\log d=H-8\log H-2u,\qquad\ell_B=\frac{C_yH}{d}=\frac{C_yL}{H^7e^{2u}},\qquad a_H=H^{1/4}. \tag*{(114)} b H = − log d = H − 8 log H − 2 u , ℓ B = d C y H = H 7 e 2 u C y L , a H = H 1/4 . (114) Consider the disks ∣ w − t ∣ ≤ .9 |w-t|\le.9 ∣ w − t ∣ ≤ .9 a = e − w a=e^{-w} a = e − w ℜ a ≥ c + e − t \Re a\ge c_+e^{-t} ℜ a ≥ c + e − t c + = e − .9 cos ( .9 ) > 0 c_+=e^{-.9}\cos(.9)>0 c + = e − .9 cos ( .9 ) > 0 ∣ a ∣ < 3 / 2 |a|<3/2 ∣ a ∣ < 3/2 t ≥ log 2 t\ge\log2 t ≥ log 2 T ∗ = 1 T_*=1 T ∗ = 1 log 2 ≤ t ≤ b H \log2\le t\le b_H log 2 ≤ t ≤ b H t ≥ t 0 = log ( 16 e .9 / τ ) t\ge t_0=\log(16e^{.9}/\tau) t ≥ t 0 = log ( 16 e .9 / τ ) T ∗ = τ / 4 T_*=\tau/4 T ∗ = τ /4
L ℓ B = H 7 e 2 u C y , L / ℓ B H ⟶ ∞ . \frac{L}{\ell_B}=\frac{H^7e^{2u}}{C_y},\qquad\frac{L/\ell_B}{H}\longrightarrow\infty. ℓ B L = C y H 7 e 2 u , H L / ℓ B ⟶ ∞. This pays any fixed power of L L L L 5 L^5 L 5 (101) at fixed height. The anchor line segment is covered by O ( ℓ B + 1 ) O(\ell_B+1) O ( ℓ B + 1 ) 1 / ( 2 ℓ B ) 1/(2\ell_B) 1/ ( 2 ℓ B ) 1 / ℓ B 1/\ell_B 1/ ℓ B (103) , finite-moment transfer, and Cauchy’s formula show that
P ( sup ∣ t ∣ ≤ T ∗ ∣ P ( 1 / 2 + a 0 + i t ) ∣ > L e 4 H ) ≪ e − 7 H . \mathbb{P}\left(\sup_{|t|\le T_*}|P(1/2+a_0+it)|>Le^{4H}\right)\ll e^{-7H}. P ( ∣ t ∣ ≤ T ∗ sup ∣ P ( 1/2 + a 0 + i t ) ∣ > L e 4 H ) ≪ e − 7 H . Indeed the second moment costs O ( ( ℓ B + 1 ) ℓ B 2 ) O((\ell_B+1)\ell_B^2) O (( ℓ B + 1 ) ℓ B 2 ) L 2 e 8 H L^2e^{8H} L 2 e 8 H ℓ B ≤ L \ell_B\le L ℓ B ≤ L E > 0 E>0 E > 0 C y C_y C y
sup t , w ∣ G D ( w ) − e − w P ( 1 / 2 + e − w ) ∣ ≤ e − E H . (115) \sup_{t,w}|G_D(w)-e^{-w}P(1/2+e^{-w})|\le e^{-EH}. \tag*{(115)} t , w sup ∣ G D ( w ) − e − w P ( 1/2 + e − w ) ∣ ≤ e − E H . (115) In the starred case the exponent in the kernel is halved, which is allowed in the choice of C y C_y C y O ( e − c ℜ ( a ) ℓ B ) O(e^{-c\Re(a)\ell_B}) O ( e − c ℜ ( a ) ℓ B ) L 2 L^2 L 2 q q q q ℓ B = o ( L ) q\ell_B=o(L) q ℓ B = o ( L )
In the Hecke family the fixed interval [ log 2 , t 0 + 2 ] [\log2,t_0+2] [ log 2 , t 0 + 2 ] a ∗ > 0 a_*>0 a ∗ > 0 ℜ ( e − w ) \Re(e^{-w}) ℜ ( e − w ) α = a ∗ / 2 \alpha=a_*/2 α = a ∗ /2 (89) . Outside probability O ( X − c ) O(X^{-c}) O ( X − c ) β ≥ α \beta\ge\alpha β ≥ α Y = X θ α Y=X^{\theta\alpha} Y = X θ α ℓ = C y H \ell=C_yH ℓ = C y H β < α \beta<\alpha β < α C ℓ − 2 L e − a ∗ ℓ / 4 C\ell^{-2}Le^{-a_*\ell/4} C ℓ − 2 L e − a ∗ ℓ /4 (97) and the negative-real kernel bound. The others contribute at most C ℓ − 2 e 7 ℓ / 4 Y − 2 log ( Q Y ) = O ( X − c ) C\ell^{-2}e^{7\ell/4}Y^{-2}\log(QY)=O(X^{-c}) C ℓ − 2 e 7 ℓ /4 Y − 2 log ( Q Y ) = O ( X − c ) (115) and its moment consequences hold on these finitely many disks as well, with constants depending on t 0 t_0 t 0
We next control the lower tails of their small-disk maxima. Write m j = max ∣ w − t j ∣ ≤ 1 / 8 ∣ G D ( w ) ∣ m_j=\max_{|w-t_j|\le1/8}|G_D(w)| m j = max ∣ w − t j ∣ ≤ 1/8 ∣ G D ( w ) ∣ v j = log + ( 1 / m j ) v_j=\log^+(1/m_j) v j = log + ( 1/ m j )
g n ( x ) = exp ( 1 − e 2 n x 2 / 4 ) , 1 { ∣ x ∣ ≤ e − n } ≤ g n ( x ) . g_n(x)=\exp(1-e^{2n}x^2/4),\qquad\mathbf{1}_{\{|x|\le e^{-n}\}}\le g_n(x). g n ( x ) = exp ( 1 − e 2 n x 2 /4 ) , 1 { ∣ x ∣ ≤ e − n } ≤ g n ( x ) . Its Fourier transform has bounded normalized L 1 L^1 L 1 ∣ v ∣ ≤ C E H e n |v|\le C_E\sqrt{H}e^n ∣ v ∣ ≤ C E H e n O ( e − E H ) O(e^{-EH}) O ( e − E H ) (105) gives E g n ( G E ( t ) ) ≪ e − a 0 n \mathbb{E}g_n(G_E(t))\ll e^{-a_0n} E g n ( G E ( t )) ≪ e − a 0 n m = [ H 4 e 2 u ] m=[H^4e^{2u}] m = [ H 4 e 2 u ] (104) , for every n ≤ u n\le u n ≤ u
∣ v ∣ m = O ( H − 3 / 2 ) , m ℓ B L = O ( H − 3 ) , m ( u + log H ) = o ( L ) . (116) \frac{|v|}{\sqrt{m}}=O(H^{-3/2}),\qquad\frac{m\ell_B}{L}=O(H^{-3}),\qquad m(u+\log H)=o(L). \tag*{(116)} m ∣ v ∣ = O ( H − 3/2 ) , L m ℓ B = O ( H − 3 ) , m ( u + log H ) = o ( L ) . (116) Thus the finite Taylor error and the arithmetic norm error are smaller than every fixed negative power of e H e^H e H O ( e n ) O(e^n) O ( e n ) (115) and in the random tail is also absorbed. Positivity permits us to drop the good-set indicator when bounding the polynomial majorant, giving
P ( g o o d , v j ≥ n ) ≪ e − a 0 n + O ( e − E H ) , 1 ≤ n ≤ u . (117) \mathbb{P}(\mathrm{good},v_j\ge n)\ll e^{-a_0n}+O(e^{-EH}),\qquad1\le n\le u. \tag*{(117)} P ( good , v j ≥ n ) ≪ e − a 0 n + O ( e − E H ) , 1 ≤ n ≤ u . (117) For the finitely many Hecke front disks, the same calculation uses m ℓ / L = O ( H 5 e 2 u − H ) → 0 m\ell/L = O(H^5e^{2u-H}) \to0 m ℓ / L = O ( H 5 e 2 u − H ) → 0 u u u O ( H e − a 0 u ) + o ( 1 ) = o ( 1 ) O(He^{-a_0u}) + o(1) = o(1) O ( H e − a 0 u ) + o ( 1 ) = o ( 1 ) (117) gives uniformly bounded restricted fourth moments of v j v_j v j
E [ 1 r e t a i n e d N j 4 ] ≪ 1 , E [ 1 r e t a i n e d sup [ t j − 1 / 4 , t j + 1 / 4 ] ∣ G D ′ ′ ∣ 8 ] ≪ 1 , (118) \mathbb{E}[\mathbf{1}_{\mathrm{retained}}N_j^4]\ll1,\qquad\mathbb{E}\left[\mathbf{1}_{\mathrm{retained}}\sup_{[t_j-1/4,t_j+1/4]}|G_D''|^8\right]\ll1, \tag*{(118)} E [ 1 retained N j 4 ] ≪ 1 , E [ 1 retained [ t j − 1/4 , t j + 1/4 ] sup ∣ G D ′′ ∣ 8 ] ≪ 1 , (118) where N j N_j N j O P ( 1 ) O_{\mathbb{P}}(1) O P ( 1 )
The weight model requires a separate early interval because the first bound in (106) grows with t t t [ log 2 , a H + 2 ] [\log2,a_H+2] [ log 2 , a H + 2 ] d F = e − a H − 3 d_F=e^{-a_H-3} d F = e − a H − 3 ℓ F = C y H / d F \ell_F=C_yH/d_F ℓ F = C y H / d F v F = 4 a H v_F=4a_H v F = 4 a H m = ⌈ H 4 e 2 v F ⌉ m=\lceil H^4e^{2v_F}\rceil m = ⌈ H 4 e 2 v F ⌉ m ℓ F / L = O ( H 5 e 9 a H + 3 − H ) → 0 m\ell_F/L=O(H^5e^{9a_H+3-H})\to0 m ℓ F / L = O ( H 5 e 9 a H + 3 − H ) → 0 (116) . The first estimate in (106) gives
P ( g o o d , v j ≥ n ) ≪ min ( 1 , e t j − n ) + e − E H . \mathbb{P}(\mathrm{good},v_j\ge n)\ll\min(1,e^{t_j-n})+e^{-EH}. P ( good , v j ≥ n ) ≪ min ( 1 , e t j − n ) + e − E H . The union of the early cap failures is O ( a H e − 3 a H + 3 ) + o ( 1 ) O(a_He^{-3a_H+3})+o(1) O ( a H e − 3 a H + 3 ) + o ( 1 ) O ( 1 + t j ) O(1+t_j) O ( 1 + t j ) O ( a H 2 ) = O ( H 1 / 2 ) = o ( H ) O(a_H^2)=O(H^{1/2})=o(H) O ( a H 2 ) = O ( H 1/2 ) = o ( H )
On the late interval [ a H − 2 , b H − 2 ] [a_H-2,b_H-2] [ a H − 2 , b H − 2 ] ℓ B \ell_B ℓ B u u u (106) gives instead P ( g o o d , v j ≥ n ) ≪ e − n + e − t j / 4 + e − E H \mathbb{P}(\mathrm{good},v_j\ge n)\ll e^{-n+e^{-t_j/4}+e^{-EH}} P ( good , v j ≥ n ) ≪ e − n + e − t j /4 + e − E H O ( H ( e − u + e − a H / 4 + e − E H ) ) = o ( 1 ) O(H(e^{-u}+e^{-a_H/4}+e^{-EH}))=o(1) O ( H ( e − u + e − a H /4 + e − E H )) = o ( 1 ) C + C u 4 ( e − a H / 4 + e − E H ) = O ( 1 ) C+Cu^4(e^{-a_H/4}+e^{-EH})=O(1) C + C u 4 ( e − a H /4 + e − E H ) = O ( 1 ) (118) holds on the late interval also in the weight family. No uniform negative moment for the weight model has been assumed.
Proposition 5.11. For each family and each fixed η > 0 \eta>0 η > 0 o η ( 1 ) o_\eta(1) o η ( 1 ) [ 1 / 2 + e 2 d , 1 ] [1/2+e^{2d},1] [ 1/2 + e 2 d , 1 ] ( c ∗ − η ) H (c_*-\eta)H ( c ∗ − η ) H ( c ∗ + η ) H (c_*+\eta)H ( c ∗ + η ) H
Proof. Work on the retained set just constructed, putting the real process equal to 1 1 1 a H a_H a H o P ( H ) o_{\mathbb{P}}(H) o P ( H ) (118) and O ( a H ) O(a_H) O ( a H )
For at most four time points in [ a H , b H − 2 ] [a_H,b_H-2] [ a H , b H − 2 ] m = ⌈ H 4 ⌉ m=\lceil H^4\rceil m = ⌈ H 4 ⌉ H H H
G ( t ) = μ + 1 2 g ( t ) , E g ( t ) g ( s ) = sech 2 ( ( t − s ) / 2 ) . \mathcal{G}(t)=\mu+\frac{1}{2}g(t),\qquad\mathbb{E}g(t)g(s)=\operatorname{sech}^{2}((t-s)/2). G ( t ) = μ + 2 1 g ( t ) , E g ( t ) g ( s ) = sech 2 (( t − s ) /2 ) . The budgets are H / m = H − 1 H/\sqrt{m}=H^{-1} H / m = H − 1 m ℓ B / L = O ( H − 3 e − 2 u ) m\ell_B/L=O(H^{-3}e^{-2u}) m ℓ B / L = O ( H − 3 e − 2 u ) K K K O ( K − 2 ) O(K^{-2}) O ( K − 2 ) ϵ \epsilon ϵ O ( ϵ ) + o ( 1 ) O(\epsilon)+o(1) O ( ϵ ) + o ( 1 ) X X X k k k K → ∞ K\to\infty K → ∞ ϵ → 0 \epsilon\to0 ϵ → 0
Fix a small mesh h > 0 h>0 h > 0 S h S_h S h
E S h H ⟶ p ( h ) h , p ( h ) = P ( G ( 0 ) G ( h ) < 0 ) . (119) \frac{\mathbb{E}S_h}{H}\longrightarrow\frac{p(h)}{h},\qquad p(h)=\mathbb{P}(\mathcal{G}(0)\mathcal{G}(h)<0). \tag*{(119)} H E S h ⟶ h p ( h ) , p ( h ) = P ( G ( 0 ) G ( h ) < 0 ) . (119) For pairs of cells separated by at most R R R O h ( H ( R + 1 ) ) O_h(H(R+1)) O h ( H ( R + 1 )) O ( e − R ) O(e^{-R}) O ( e − R ) O ( ε − 2 e − R ) O(\varepsilon^{-2}e^{-R}) O ( ε − 2 e − R ) O ( ε ) O(\varepsilon) O ( ε ) ε = e − R / 3 \varepsilon=e^{-R/3} ε = e − R /3 O ( e − R / 3 ) O(e^{-R/3}) O ( e − R /3 )
lim sup X → ∞ Var ( S h / H ) ≪ h e − R / 3 . \limsup_{X\to\infty}\operatorname{Var}(S_h/H)\ll_h e^{-R/3}. X → ∞ lim sup Var ( S h / H ) ≪ h e − R /3 . Let R → ∞ R\to\infty R → ∞ k k k S h / H − p ( h ) / h → 0 S_h/H-p(h)/h\to0 S h / H − p ( h ) / h → 0
It remains to recover all multiplicities from mesh signs. If a cell of length h h h
∣ G D ( t l e f t ) ∣ ≤ h 2 sup I ∣ G D ′ ′ ∣ . |G_D(t_{\mathrm{left}})|\le h^2\sup_I|G_D''|. ∣ G D ( t left ) ∣ ≤ h 2 I sup ∣ G D ′′ ∣. Set K = h − 1 / 4 K=h^{-1/4} K = h − 1/4 (118) imply
P ( r e t a i n e d , N I ≥ 2 ) ≪ h 7 / 4 + h 2 + o ( 1 ) . \mathbb{P}(\mathrm{retained},N_I\ge2)\ll h^{7/4}+h^2+o(1). P ( retained , N I ≥ 2 ) ≪ h 7/4 + h 2 + o ( 1 ) . Hölder and the first bound in (118) give E [ 1 r e t a i n e d N I 1 N I ≥ 2 ] ≪ h 21 / 16 + o ( 1 ) \mathbb{E}[\mathbf{1}_{\mathrm{retained}}N_I\mathbf{1}_{N_I\ge2}]\ll h^{21/16}+o(1) E [ 1 retained N I 1 N I ≥ 2 ] ≪ h 21/16 + o ( 1 ) O ( H / h ) O(H/h) O ( H / h ) C H h 5 / 16 + o ( H ) CHh^{5/16}+o(H) C H h 5/16 + o ( H ) O P ( 1 ) O_{\mathbb{P}}(1) O P ( 1 ) H H H S h / H S_h/H S h / H C h 5 / 16 + o ( 1 ) Ch^{5/16}+o(1) C h 5/16 + o ( 1 )
Finally the Gaussian calculation (20) , with the parameters of (102) , gives p ( h ) = c ∗ h + O ( h 3 ) p(h)=c_*h+O(h^3) p ( h ) = c ∗ h + O ( h 3 ) μ = 1 / 2 \mu=1/2 μ = 1/2 − 1 / 2 -1/2 − 1/2 h h h η \eta η b H − 2 b_H-2 b H − 2 e 2 d e^{2d} e 2 d
Assembly and normalization
The three physical intervals
[ 1 / 2 , 1 / 2 + Δ ] , [ 1 / 2 + Δ , 1 / 2 + e 5 d ] , [ 1 / 2 + e 2 d , 1 ] [1/2,1/2+\Delta],\qquad[1/2+\Delta,1/2+e^5d],\qquad[1/2+e^2d,1] [ 1/2 , 1/2 + Δ ] , [ 1/2 + Δ , 1/2 + e 5 d ] , [ 1/2 + e 2 d , 1 ] cover [ 1 / 2 , 1 ] [1/2,1] [ 1/2 , 1 ] o P ( H ) o_{\mathbb{P}}(H) o P ( H ) c ∗ H + o P ( H ) c_*H+o_{\mathbb{P}}(H) c ∗ H + o P ( H )
R d i s t ( L ′ ) = c ∗ H + o P ( H ) , R m u l t ( L ′ ) = c ∗ H + o P ( H ) . R^{\mathrm{dist}}(L')=c_*H+o_{\mathbb{P}}(H),\qquad R^{\mathrm{mult}}(L')=c_*H+o_{\mathbb{P}}(H). R dist ( L ′ ) = c ∗ H + o P ( H ) , R mult ( L ′ ) = c ∗ H + o P ( H ) . The deterministic multiplicity/sign inequality of Lemma 4.3 is
2 R d i s t ( L ′ ) − R m u l t ( L ′ ) − 2 ≤ S − ( L ′ ) ≤ R d i s t ( L ′ ) . 2R^{\mathrm{dist}}(L')-R^{\mathrm{mult}}(L')-2\le S^-(L')\le R^{\mathrm{dist}}(L'). 2 R dist ( L ′ ) − R mult ( L ′ ) − 2 ≤ S − ( L ′ ) ≤ R dist ( L ′ ) . The subtraction of two allows for the closed endpoints, and gives the same asymptotic for strict changes of sign on the open interval.
For the rational family discard ∣ D ∣ ≤ X |D|\le\sqrt{X} ∣ D ∣ ≤ X O ( X − 1 / 2 ) O(X^{-1/2}) O ( X − 1/2 ) (62) . On the remaining family, H − log 2 ≤ log log ∣ D ∣ ≤ H H-\log2\le\log\log|D|\le H H − log 2 ≤ log log ∣ D ∣ ≤ H H H H X X X Theorem 5.1 with the full cumulative denominator, not a conditional dyadic denominator. For the Gaussian family 128 X < Q d ≤ 256 X 128X<Q_d\le256X 128 X < Q d ≤ 256 X log log Q d = H + O ( 1 / L ) \log\log Q_d=H+O(1/L) log log Q d = H + O ( 1/ L ) ∣ D X ∣ ≍ X |\mathcal{D}_X|\asymp X ∣ D X ∣ ≍ X (64) preserve every probability, proving Theorem 5.2 also for distinct characters. The cumulative cutoff follows by discarding N d ≤ 2 − J Y N_d\le2^{-J}Y N d ≤ 2 − J Y J J J Y → ∞ Y\to\infty Y → ∞ J → ∞ J\to\infty J → ∞ L = log k L=\log k L = log k log log ( k 2 ) = H + log 2 \log\log(k^2)=H+\log2 log log ( k 2 ) = H + log 2 ( 2 π ) − s Γ ( s + ( k − 1 ) / 2 ) (2\pi)^{-s}\Gamma(s+(k-1)/2) ( 2 π ) − s Γ ( s + ( k − 1 ) /2 ) i k i^k i k Theorem 5.3 along each even residue class modulo four, and hence along all even weights.
Three arithmetic families under GRH
We verify the arithmetic and analytic hypotheses of the preceding transfer results for three real families. All averages in this section are counting averages. In particular, no harmonic weight, automorphism weight, or choice of one member of a conjugate pair is implicit. We subsequently treat complex cubic characters separately; their real zeros are not sign changes of a real-valued process.
For a real L L L
A F ( p j ) = the coefficient of ( log p ) p − j s in − L F ′ L F ( s ) , G F ( t ) = e − t − L F ′ L F ( 1 2 + e − t ) . A_F(p^j)=\text{the coefficient of }(\log p)p^{-js}\text{ in }-\frac{L'_F}{L_F}(s),\qquad G_F(t)=e^{-t}\frac{-L'_F}{L_F}\left(\frac{1}{2}+e^{-t}\right). A F ( p j ) = the coefficient of ( log p ) p − j s in − L F L F ′ ( s ) , G F ( t ) = e − t L F − L F ′ ( 2 1 + e − t ) . A word W = ∏ i = 1 r A F ( p i j i ) W=\prod_{i=1}^{r}A_F(p_i^{j_i}) W = ∏ i = 1 r A F ( p i j i ) r r r n ( W ) = ∏ i p i j i n(W)=\prod_i p_i^{j_i} n ( W ) = ∏ i p i j i
∣ E F ( Y ) W − E W E ∣ ≤ C Y − κ C 0 r n ( W ) A . (120) \left|\mathbb{E}_{F(Y)}W-\mathbb{E}_W\mathbb{E}\right|\le CY^{-\kappa}C_0^r n(W)^A. \tag*{(120)} E F ( Y ) W − E W E ≤ C Y − κ C 0 r n ( W ) A . (120) For a formal polynomial P = ∑ W c W W P=\sum_W c_W W P = ∑ W c W W
∥ P ∥ ∗ = ∑ W ∣ c W ∣ C 0 ∣ W ∣ n ( W ) A (121) \lVert P\rVert_*=\sum_W |c_W|C_0^{|W|}n(W)^A \tag*{(121)} ∥ P ∥ ∗ = W ∑ ∣ c W ∣ C 0 ∣ W ∣ n ( W ) A (121) is submultiplicative, and the error in its arithmetic mean is at most C Y − κ ∥ P ∥ ∗ CY^{-\kappa}\lVert P\rVert_* C Y − κ ∥ P ∥ ∗ Proposition 2.16 .
The local parameters needed for the limiting real process are as follows. Here Y Y Y Q F Q_F Q F g g g sech 2 ( ( t − s ) / 2 ) \operatorname{sech}^2((t-s)/2) sech 2 (( t − s ) /2 )
Table 1 (PDF p. 54).
The last exponent is expressed in the conductor parameter Y Y Y Y 2 Y^2 Y 2 1 / 8 1/8 1/8
Fixed modular forms and all quadratic discriminants
Let f f f k ≥ 2 k\ge2 k ≥ 2 N N N
f ( z ) = ∑ n ≥ 1 λ f ( n ) n ( k − 1 ) / 2 e 2 π i n z . f(z)=\sum_{n\ge1}\lambda_f(n)n^{(k-1)/2}e^{2\pi inz}. f ( z ) = n ≥ 1 ∑ λ f ( n ) n ( k − 1 ) /2 e 2 π in z . The coefficients are real. No non-CM hypothesis is imposed. At p ∤ N p\nmid N p ∤ N α p + β p = λ f ( p ) \alpha_p+\beta_p=\lambda_f(p) α p + β p = λ f ( p ) α p β p = 1 \alpha_p\beta_p=1 α p β p = 1 [12 Théorème 8.2] gives ∣ α p ∣ = ∣ β p ∣ = 1 |\alpha_p|=|\beta_p|=1 ∣ α p ∣ = ∣ β p ∣ = 1
b f ( p j ) = α p j + β p j , b f ( p ) = λ f ( p ) , b f ( p 2 ) = λ f ( p ) 2 − 2. (122) b_f(p^j)=\alpha_p^j+\beta_p^j,\qquad b_f(p)=\lambda_f(p),\qquad b_f(p^2)=\lambda_f(p)^2-2. \tag*{(122)} b f ( p j ) = α p j + β p j , b f ( p ) = λ f ( p ) , b f ( p 2 ) = λ f ( p ) 2 − 2. (122) The last expression is not the Hecke coefficient λ f ( p 2 ) \lambda_f(p^2) λ f ( p 2 ) N N N ( 1 − λ f ( p ) p − s ) − 1 (1-\lambda_f(p)p^{-s})^{-1} ( 1 − λ f ( p ) p − s ) − 1
λ f ( p ) = 0 ( p 2 ∣ N ) , ∣ λ f ( p ) ∣ = p − 1 / 2 ( p ∥ N ) , (123) \lambda_f(p)=0\ (p^2\mid N),\qquad|\lambda_f(p)|=p^{-1/2}\ (p\|N), \tag*{(123)} λ f ( p ) = 0 ( p 2 ∣ N ) , ∣ λ f ( p ) ∣ = p − 1/2 ( p ∥ N ) , (123) by [1 Theorem 3] . Thus all the finitely many bad-prime blocks and their derivatives are bounded on a fixed neighborhood of the real interval [ 1 / 2 , 1 ] [1/2,1] [ 1/2 , 1 ]
Lemma 6.1 (Quadratic averages in a fixed progression). Fix M M M 8 N 8N 8 N a a a M M M a ≡ 1 a\equiv1 a ≡ 1 η ∈ { 1 , − 1 } \eta\in\{1,-1\} η ∈ { 1 , − 1 } D ( Y ) D(Y) D ( Y ) Y < ∣ D ∣ ≤ 2 Y Y<|D|\le2Y Y < ∣ D ∣ ≤ 2 Y η D > 0 \eta D>0 η D > 0 D ≡ a D\equiv a D ≡ a M M M p ∤ M p\nmid M p ∤ M X p X_p X p
P ( X p = 0 ) = 1 p + 1 , P ( X p = 1 ) = P ( X p = − 1 ) = p 2 ( p + 1 ) . \mathbb{P}(X_p=0)=\frac{1}{p+1},\qquad\mathbb{P}(X_p=1)=\mathbb{P}(X_p=-1)=\frac{p}{2(p+1)}. P ( X p = 0 ) = p + 1 1 , P ( X p = 1 ) = P ( X p = − 1 ) = 2 ( p + 1 ) p . At p ∣ M p\mid M p ∣ M X p = χ D ( p ) X_p=\chi_D(p) X p = χ D ( p ) n n n
E D ∈ D ( Y ) χ D ( n ) = E X ( n ) + O M ( Y − 1 / 2 n ) , X ( n ) = ∏ p X p v p ( n ) . (124) \mathbb{E}_{D\in D(Y)}\chi_D(n)=\mathbb{E}X(n)+O_M(Y^{-1/2}n),\qquad X(n)=\prod_p X_p^{v_p(n)}. \tag*{(124)} E D ∈ D ( Y ) χ D ( n ) = E X ( n ) + O M ( Y − 1/2 n ) , X ( n ) = p ∏ X p v p ( n ) . (124) Also ∣ D ( Y ) ∣ = c M Y + O M ( Y ) |D(Y)|=c_MY+O_M(\sqrt{Y}) ∣ D ( Y ) ∣ = c M Y + O M ( Y ) c M = M − 1 ∏ p ∤ M ( 1 − p − 2 ) > 0 c_M=M^{-1}\prod_{p\nmid M}(1-p^{-2})>0 c M = M − 1 ∏ p ∤ M ( 1 − p − 2 ) > 0
Proof. Write D = η d D=\eta d D = η d d > 0 d>0 d > 0 μ 2 ( d ) = ∑ h 2 ∣ d μ ( h ) \mu^2(d)=\sum_{h^2\mid d}\mu(h) μ 2 ( d ) = ∑ h 2 ∣ d μ ( h ) ( h , M ) = 1 (h,M)=1 ( h , M ) = 1 h − 2 h^{-2} h − 2 O M ( Y ) O_M(\sqrt{Y}) O M ( Y ) n n n M M M n 0 n_0 n 0
If n 0 n_0 n 0 ( ⋅ / n 0 ) (\cdot/n_0) ( ⋅ / n 0 ) n 0 n_0 n 0 ( h , n 0 ) = 1 (h,n_0)=1 ( h , n 0 ) = 1 h 2 r ≡ η a h^2r\equiv\eta a h 2 r ≡ η a M M M n 0 n_0 n 0 O ( n 0 ) O(n_0) O ( n 0 ) ( h , n 0 ) > 1 (h,n_0)>1 ( h , n 0 ) > 1 h ≤ 2 Y h\le\sqrt{2Y} h ≤ 2 Y O M ( n 0 Y ) O_M(n_0\sqrt{Y}) O M ( n 0 Y )
If n 0 n_0 n 0 1 ( d , n 0 ) = 1 \mathbf{1}_{(d,n_0)=1} 1 ( d , n 0 ) = 1 h h h M n 0 Mn_0 M n 0 r r r
Y M h 2 ∏ p ∣ n 0 ( 1 − p − 1 ) \frac{Y}{Mh^2}\prod_{p\mid n_0}(1-p^{-1}) M h 2 Y p ∣ n 0 ∏ ( 1 − p − 1 ) and an error O ( 2 ω ( n 0 ) ) O(2^{\omega(n_0)}) O ( 2 ω ( n 0 ) ) h h h c M Y c_MY c M Y
∏ p ∣ n 0 1 − p − 1 1 − p − 2 = ∏ p ∣ n 0 p p + 1 . \prod_{p\mid n_0}\frac{1-p^{-1}}{1-p^{-2}}=\prod_{p\mid n_0}\frac{p}{p+1}. p ∣ n 0 ∏ 1 − p − 2 1 − p − 1 = p ∣ n 0 ∏ p + 1 p . The error is O M ( Y − 1 / 2 2 ω ( n 0 ) ) O_M(Y^{-1/2}2^{\omega(n_0)}) O M ( Y − 1/2 2 ω ( n 0 ) ) (124) . The signs introduced by D = η d D=\eta d D = η d □ \square □
For ( D , N ) = 1 (D,N)=1 ( D , N ) = 1 Q D = N D 2 Q_D=ND^2 Q D = N D 2
Λ D ( s ) = ( Q D 2 π ) s Γ ( s + ( k − 1 ) / 2 ) L ( s , f ⊗ χ D ) , \Lambda_D(s)=\left(\frac{\sqrt{Q_D}}{2\pi}\right)^s\Gamma(s+(k-1)/2)L(s,f\otimes\chi_D), Λ D ( s ) = ( 2 π Q D ) s Γ ( s + ( k − 1 ) /2 ) L ( s , f ⊗ χ D ) , Λ D ( s ) = ϵ f χ D ( − N ) Λ D ( 1 − s ) . (125) \Lambda_D(s)=\epsilon_f\chi_D(-N)\Lambda_D(1-s). \tag*{(125)} Λ D ( s ) = ϵ f χ D ( − N ) Λ D ( 1 − s ) . (125) These are entire functions of order one, with the usual fixed-degree polynomial growth in vertical strips. Both root-number signs are allowed; a forced central zero is dealt with by the central zero estimate, and is not removed from the family. The primitive twisting statements used here are [2 §3] ; see also the normalization in [24 pp. 1–2] .
Lemma 6.2 (The modular local process). The model of Lemma 6.1 satisfies, with a = e − t a=e^{-t} a = e − t
E G E ( t ) = − 1 2 + O f , M ( a ) , (126) \mathbb{E}G_E(t)=-\frac{1}{2}+O_{f,M}(a), \tag*{(126)} E G E ( t ) = − 2 1 + O f , M ( a ) , (126) Cov ( G E ( − log a ) , G E ( − log b ) ) = a b ( a + b ) 2 + O f , M ( a b ) . (127) \operatorname{Cov}(G_E(-\log a),G_E(-\log b))=\frac{ab}{(a+b)^2}+O_{f,M}(ab). \tag*{(127)} Cov ( G E ( − log a ) , G E ( − log b )) = ( a + b ) 2 ab + O f , M ( ab ) . (127) The estimates extend to fixed closed subsectors of ℜ a > 0 \Re a>0 ℜ a > 0 in their analytic form. Normalized local analytic moments of every fixed order are bounded, with constants allowed to depend on the fixed modulus M M M For p ∤ M p\nmid M p ∤ M the full local blocks have a two-atom difference sufficient for the uniform small-value estimate of Lemma 2.5 .
Proof. The Rankin–Selberg prime number theorem of [22 Theorem 2.3] , applied to the self-contragredient representation of f f f
∑ n ≤ x Λ ( n ) ∣ a f ( n ) ∣ 2 = x + O f ( x e − c f log x ) . \sum_{n\le x}\Lambda(n)|a_f(n)|^2=x+O_f\left(xe^{-c_f\sqrt{\log x}}\right). n ≤ x ∑ Λ ( n ) ∣ a f ( n ) ∣ 2 = x + O f ( x e − c f l o g x ) . Deligne’s bound removes the good prime powers of exponent at least two with error O ( x log 2 x ) O(\sqrt{x}\log^2 x) O ( x log 2 x ) (123) handles the finitely many remaining primes. Consequently
∑ p ≤ x p ∤ M λ f ( p ) 2 log p = x + O f ( x e − c f log x ) . (128) \sum_{\substack{p\le x\\p\nmid M}}\lambda_f(p)^2\log p=x+O_f\left(xe^{-c_f\sqrt{\log x}}\right). \tag*{(128)} p ≤ x p ∤ M ∑ λ f ( p ) 2 log p = x + O f ( x e − c f l o g x ) . (128) This argument includes CM forms. Partial summation, and the ordinary prime number theorem, now give
∑ p λ f ( p ) 2 log p p − 1 − z = z − 1 + O f ( 1 ) , ∑ p λ f ( p ) 2 ( log p ) 2 p − 1 − z = z − 2 + O f ( 1 ) . \sum_p\lambda_f(p)^2\log p\,p^{-1-z}=z^{-1}+O_f(1),\qquad
\sum_p\lambda_f(p)^2(\log p)^2p^{-1-z}=z^{-2}+O_f(1). p ∑ λ f ( p ) 2 log p p − 1 − z = z − 1 + O f ( 1 ) , p ∑ λ f ( p ) 2 ( log p ) 2 p − 1 − z = z − 2 + O f ( 1 ) . The second formula follows either by differentiating the convergent remainder integral or by a second partial summation; both are uniform in a fixed closed subsector of ℜ z > 0 \Re z>0 ℜ z > 0
For p ∤ M p\nmid M p ∤ M X p X_p X p p / ( p + 1 ) p/(p+1) p / ( p + 1 ) p ∣ M p\mid M p ∣ M O f , M ( a ) O_{f,M}(a) O f , M ( a ) j = 2 j=2 j = 2 M M M (122) , therefore have mean − 1 / ( 2 a ) + O f , M ( 1 ) -1/(2a)+O_{f,M}(1) − 1/ ( 2 a ) + O f , M ( 1 ) a a a j ≥ 4 j\ge4 j ≥ 4 j ≥ 2 j\ge2 j ≥ 2 L 2 L^2 L 2 a a a O f ( ( log p ) / p ) O_f((\log p)/p) O f (( log p ) / p ) j = 1 j=1 j = 1 (128) give (127) ; covariance cross terms are summable as well. The bounded independent prime blocks give the fixed-order moment bounds by the exponential-moment inequality, followed by Cauchy’s formula on a slightly larger sectorial disc.
For clarity, the atom calculation concerns the full block, not just its first term. Put u = p − 1 / 2 − a u=p^{-1/2-a} u = p − 1/2 − a
B p ( x ) = ( log p ) λ f ( p ) x u − 2 x 2 u 2 1 − λ f ( p ) x u + x 2 u 2 . B_p(x)=(\log p)\frac{\lambda_f(p)xu-2x^2u^2}{1-\lambda_f(p)xu+x^2u^2}. B p ( x ) = ( log p ) 1 − λ f ( p ) xu + x 2 u 2 λ f ( p ) xu − 2 x 2 u 2 . Then
B p ( 1 ) − B p ( − 1 ) = 2 λ f ( p ) u ( 1 − u 2 ) log p ( 1 + u 2 ) 2 − λ f ( p ) 2 u 2 . (129) B_p(1)-B_p(-1)=\frac{2\lambda_f(p)u(1-u^2)\log p}{(1+u^2)^2-\lambda_f(p)^2u^2}. \tag*{(129)} B p ( 1 ) − B p ( − 1 ) = ( 1 + u 2 ) 2 − λ f ( p ) 2 u 2 2 λ f ( p ) u ( 1 − u 2 ) log p . (129) The factor multiplying λ f ( p ) u log p \lambda_f(p)u\log p λ f ( p ) u log p p ≥ 2 p\ge2 p ≥ 2 a > 0 a>0 a > 0 p ∤ M p\nmid M p ∤ M λ f ( p ) \lambda_f(p) λ f ( p ) (128) supplies the required weighted prime mass on dyadic intervals. It is thus unnecessary to impose a pointwise lower bound on ∣ λ f ( p ) ∣ |\lambda_f(p)| ∣ λ f ( p ) ∣
Lemma 6.1 and ∣ b f ( p j ) ∣ ≤ 2 |b_f(p^j)| \le2 ∣ b f ( p j ) ∣ ≤ 2 (120) with ( κ , A , C 0 ) = ( 1 / 2 , 1 , 2 ) (\kappa,A,C_0)=(1/2,1,2) ( κ , A , C 0 ) = ( 1/2 , 1 , 2 ) p j ≤ e 3 ℓ p^j \le e^{3\ell} p j ≤ e 3 ℓ (121) is at most e C f , M ( ℓ + 1 ) e^{C_{f,M}(\ell+1)} e C f , M ( ℓ + 1 ) 2 m 2m 2 m
( C f , M m ) 2 m + Y − 1 / 2 exp ( C f , M m ( ℓ + 1 ) ) . (130) (C_{f,M}\sqrt{m})^{2m}+Y^{-1/2}\exp(C_{f,M}m(\ell+1)). \tag*{(130)} ( C f , M m ) 2 m + Y − 1/2 exp ( C f , M m ( ℓ + 1 )) . (130) Indeed, the random estimate follows from independent bounded prime blocks, and expansion of the finite power followed by (120) proves the arithmetic estimate. This also proves the required weighted-polynomial input when the number of primes grows; there is no fixed-cylinder restriction.
Proposition 6.3 (All discriminants and coefficientwise twists). The preceding inputs apply to all signed fundamental discriminants, with L ( s , f ⊗ χ D ) L(s,f\otimes\chi_D) L ( s , f ⊗ χ D ) Q D ≍ f D 2 Q_D\asymp_f D^2 Q D ≍ f D 2
L D c o e f f ( s ) = ∑ n ≥ 1 λ f ( n ) χ D ( n ) n − s , L_D^{\mathrm{coeff}}(s)=\sum_{n\ge1}\lambda_f(n)\chi_D(n)n^{-s}, L D coeff ( s ) = n ≥ 1 ∑ λ f ( n ) χ D ( n ) n − s , using the conductor of its primitive representative as the reference conductor. The GRH assumption is imposed on the primitive twists.
Proof. Let S S S 2 N 2N 2 N p ∗ = ( − 1 ) ( p − 1 ) / 2 p p^*=(-1)^{(p-1)/2}p p ∗ = ( − 1 ) ( p − 1 ) /2 p p p p
D = D 2 ∏ p ∣ D p o d d p ∗ , D 2 ∈ { 1 , − 4 , 8 , − 8 } . D=D_2\prod_{\substack{p\mid D\\p\ \mathrm{odd}}}p^*,\qquad D_2\in\{1,-4,8,-8\}. D = D 2 p ∣ D p odd ∏ p ∗ , D 2 ∈ { 1 , − 4 , 8 , − 8 } . Separate the factors supported on S S S D = D S d D=D_Sd D = D S d χ D = θ χ d \chi_D=\theta\chi_d χ D = θ χ d θ = χ D S \theta=\chi_{D_S} θ = χ D S d d d S S S [2 Theorem 3.2] produce a fixed primitive form f θ f_\theta f θ N θ N_\theta N θ S S S [1 Theorem 4] , since its Hecke eigenvalues away from S S S λ f ( p ) θ ( p ) \lambda_f(p)\theta(p) λ f ( p ) θ ( p ) χ d \chi_d χ d
L ( s , f ⊗ χ D ) = L ( s , f θ ⊗ χ d ) , Q D = N θ d 2 = N θ ∣ D S ∣ 2 D 2 . (131) L(s,f\otimes\chi_D)=L(s,f_\theta\otimes\chi_d),\qquad Q_D=N_\theta d^2=\frac{N_\theta}{|D_S|^2}D^2. \tag*{(131)} L ( s , f ⊗ χ D ) = L ( s , f θ ⊗ χ d ) , Q D = N θ d 2 = ∣ D S ∣ 2 N θ D 2 . (131) Partition the remaining d d d 8 ∏ p ∈ S p 8\prod_{p\in S}p 8 ∏ p ∈ S p f θ f_\theta f θ
The coefficientwise twist differs at only the common ramified primes:
L D c o e f f ( s ) = L ( s , f θ ⊗ χ d ) E D ( s ) , E D ( s ) = ∏ p ∣ ( D , N ) L p ( s , f θ ⊗ χ d ) − 1 . (132) L_D^{\mathrm{coeff}}(s)=L(s,f_\theta\otimes\chi_d)E_D(s),\qquad E_D(s)=\prod_{p\mid(D,N)}L_p(s,f_\theta\otimes\chi_d)^{-1}. \tag*{(132)} L D coeff ( s ) = L ( s , f θ ⊗ χ d ) E D ( s ) , E D ( s ) = p ∣ ( D , N ) ∏ L p ( s , f θ ⊗ χ d ) − 1 . (132) This is also the depletion assertion of [2 Theorem 3.2] . The local inverse factors have real coefficients and all reciprocal roots of modulus at most one. After a further finite partition, E D E_D E D R s ≤ 0 \mathbb{R}s\le0 R s ≤ 0 E D ′ / E D E_D'/E_D E D ′ / E D [ 1 / 2 , 1 ] [1/2,1] [ 1/2 , 1 ] O f ( ∣ a ∣ ) O_f(|a|) O f ( ∣ a ∣ ) L D c o e f f L_D^{\mathrm{coeff}} L D coeff Proposition 4.4 to each of the finite classes.
Non-Galois cubic fields
Let F ± ( Y ) \mathcal{F}_{\pm}(Y) F ± ( Y ) K K K Y < ∣ D K ∣ ≤ 2 Y Y < |D_K| \le2Y Y < ∣ D K ∣ ≤ 2 Y
L K ( s ) = ζ K ( s ) ζ ( s ) , Q K = ∣ D K ∣ . (133) L_K(s)=\frac{\zeta_K(s)}{\zeta(s)},\qquad Q_K=|D_K|. \tag*{(133)} L K ( s ) = ζ ( s ) ζ K ( s ) , Q K = ∣ D K ∣. (133) This Artin L L L + 1 +1 + 1 Γ R ( s ) 2 \Gamma_{\mathbb{R}}(s)^2 Γ R ( s ) 2 D K > 0 D_K>0 D K > 0 Γ R ( s ) Γ R ( s + 1 ) \Gamma_{\mathbb{R}}(s)\Gamma_{\mathbb{R}}(s+1) Γ R ( s ) Γ R ( s + 1 ) D K < 0 D_K<0 D K < 0 Γ R ( s ) = π − s / 2 Γ ( s / 2 ) \Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2) Γ R ( s ) = π − s /2 Γ ( s /2 ) [25 Propositions 3.1–3.2 and (22)] . GRH here concerns L K L_K L K ζ \zeta ζ
Put D p = p 2 + p + 1 D_p=p^2+p+1 D p = p 2 + p + 1
Table 1 (PDF p. 58).
The last two rows include all their local-algebra refinements, also at 2 and 3.
Lemma 6.4 (Uniform cubic-field word averages). For the independent five-state local model in the table , (120) holds on either F ± ( Y ) \mathcal{F}_{\pm}(Y) F ± ( Y ) ( κ , A , C 0 ) = ( 1 / 6 , 1 , 10 ) (\kappa,A,C_0)=(1/6,1,10) ( κ , A , C 0 ) = ( 1/6 , 1 , 10 )
Proof. Theorem 1.3 and Table 1 of [6 ] count cubic fields with finitely many local specifications. In its notation the count up to Y Y Y
A ± Y ∏ p ∈ S C p + B ± Y 5 / 6 ∏ p ∈ S K p + O δ ( Y 2 / 3 + δ ∏ p ∈ S E p ) , (134) A_{\pm}Y\prod_{p\in S}C_p+B_{\pm}Y^{5/6}\prod_{p\in S}K_p+O_{\delta}\left(Y^{2/3+\delta}\prod_{p\in S}E_p\right), \tag*{(134)} A ± Y p ∈ S ∏ C p + B ± Y 5/6 p ∈ S ∏ K p + O δ Y 2/3 + δ p ∈ S ∏ E p , (134) where
A ± = C ± 12 ζ ( 3 ) , C + = 1 , C − = 3 , B ± = 4 K ± ζ ( 1 / 3 ) 5 Γ ( 2 / 3 ) 3 ζ ( 5 / 3 ) , K + = 1 , K − = 3 . A_{\pm}=\frac{C_{\pm}}{12\zeta(3)},\qquad C_+=1,\ C_-=3,\qquad B_{\pm}=\frac{4K_{\pm}\zeta(1/3)}{5\Gamma(2/3)^3\zeta(5/3)},\qquad K_+=1,\ K_-=\sqrt{3}. A ± = 12 ζ ( 3 ) C ± , C + = 1 , C − = 3 , B ± = 5Γ ( 2/3 ) 3 ζ ( 5/3 ) 4 K ± ζ ( 1/3 ) , K + = 1 , K − = 3 . For the five aggregate types, C p C_p C p K p K_p K p E p ≤ p 2 / 3 E_p\le p^{2/3} E p ≤ p 2/3
Subtracting the formula at 2 Y 2Y 2 Y Y Y Y 2 5 / 6 − 1 2^{5/6}-1 2 5/6 − 1 O ( Y log ( 2 Y ) ) O(\sqrt{Y}\log(2Y)) O ( Y log ( 2 Y )) 2 Y \sqrt{2Y} 2 Y 2 τ ( q ) 2\tau(q) 2 τ ( q )
If F F F S S S 5 ∣ S ∣ 5^{|S|} 5 ∣ S ∣
∣ E F ± ( Y ) F − E F ∣ ≪ δ ∥ F ∥ ∞ ( Y − 1 / 6 + Y − 1 / 3 + δ 5 ∣ S ∣ ( ∏ p ∈ S p ) 2 / 3 + Y − 1 / 2 log ( 2 Y ) ) . (135) \left|\mathbb{E}_{\mathcal{F}_{\pm}(Y)}F-\mathbb{E}F\right|\ll_{\delta}\|F\|_{\infty}\left(Y^{-1/6}+Y^{-1/3+\delta}5^{|S|}\left(\prod_{p\in S}p\right)^{2/3}+Y^{-1/2}\log(2Y)\right). \tag*{(135)} E F ± ( Y ) F − E F ≪ δ ∥ F ∥ ∞ Y − 1/6 + Y − 1/3 + δ 5 ∣ S ∣ p ∈ S ∏ p 2/3 + Y − 1/2 log ( 2 Y ) . (135) The secondary term costs ∥ F ∥ ∞ \|F\|_{\infty} ∥ F ∥ ∞ 5 ∣ S ∣ ∥ F ∥ ∞ 5^{|S|}\|F\|_{\infty} 5 ∣ S ∣ ∥ F ∥ ∞ K p K_p K p r r r ∥ F ∥ ∞ ≤ 2 r \|F\|_{\infty}\le2^r ∥ F ∥ ∞ ≤ 2 r ∣ S ∣ ≤ r |S|\le r ∣ S ∣ ≤ r ∏ p ∈ S p ≤ n ( W ) \prod_{p\in S}p\le n(W) ∏ p ∈ S p ≤ n ( W ) δ = 1 / 12 \delta=1/12 δ = 1/12 (135) proves the claim. In particular, the secondary term prevents replacing κ = 1 / 6 \kappa=1/6 κ = 1/6 (134) . □
Lemma 6.5 (The cubic-field local process). The five-state model has limiting normalized process 1 / 2 + g / 2 1/2+g/2 1/2 + g /2
Proof. Directly from the table,
E A ( p ) = p D p , E A ( p ) 2 = E A ( p 2 ) = p 2 + p D p , Var A ( p ) = 1 − 1 D p − p 2 D p 2 . (136) \mathbb{E}A(p)=\frac{p}{D_p},\qquad\mathbb{E}A(p)^2=\mathbb{E}A(p^2)=\frac{p^2+p}{D_p},\qquad\operatorname{Var} A(p)=1-\frac{1}{D_p}-\frac{p^2}{D_p^2}. \tag*{(136)} E A ( p ) = D p p , E A ( p ) 2 = E A ( p 2 ) = D p p 2 + p , Var A ( p ) = 1 − D p 1 − D p 2 p 2 . (136) More generally, E A ( p j ) = [ p 2 ( 1 2 ∣ j + 1 3 ∣ j ) + p ] / D p \mathbb{E}A(p^j)=[p^2(1_{2\mid j}+1_{3\mid j})+p]/D_p E A ( p j ) = [ p 2 ( 1 2 ∣ j + 1 3 ∣ j ) + p ] / D p p − 1 / 2 − a log p p^{-1/2-a}\log p p − 1/2 − a log p 1 + O ( p − 2 ) 1+O(p^{-2}) 1 + O ( p − 2 ) 1 / 2 1/2 1/2 1 + O ( p − 2 ) 1+O(p^{-2}) 1 + O ( p − 2 ) a b / ( a + b ) 2 + O ( a b ) ab/(a+b)^2+O(ab) ab / ( a + b ) 2 + O ( ab )
The identity and transposition atoms have probabilities at least 2 / 21 2/21 2/21 2 / 7 2/7 2/7 u = p − 1 / 2 − a u=p^{-1/2-a} u = p − 1/2 − a
2 a ( log p ) u 1 − u 2 . (137) 2a(\log p)\frac{u}{1-u^2}. \tag*{(137)} 2 a ( log p ) 1 − u 2 u . (137) It lies between 2 a ( log p ) u 2a(\log p)u 2 a ( log p ) u 4 a ( log p ) u 4a(\log p)u 4 a ( log p ) u P = ( C T log 2 ( 2 T ) ) 1 / ( 1 / 2 + a ) P=(CT\log^2(2T))^{1/(1/2+a)} P = ( C T log 2 ( 2 T ) ) 1/ ( 1/2 + a ) T = 2 + ∣ τ ∣ T=2+|\tau| T = 2 + ∣ τ ∣ τ \tau τ a a a
The word norm of a Selberg polynomial is again e C ( ℓ + 1 ) e^{C(\ell+1)} e C ( ℓ + 1 )
Oriented cubic characters: a uniform arithmetic lemma
Let C ( Y ) \mathcal{C}(Y) C ( Y ) Y Y Y C 0 ( Y ) \mathcal{C}_0(Y) C 0 ( Y ) N , N 0 N,N_0 N , N 0 ω = e 2 π i / 3 \omega=e^{2\pi i/3} ω = e 2 π i /3
Proposition 6.6 (Oriented cubic orthogonality). Let Z p Z_p Z p { 0 , 1 , ω , ω 2 } \{0,1,\omega,\omega^2\} { 0 , 1 , ω , ω 2 }
P ( Z p = 0 ) = 1 − θ p , P ( Z p = ω j ) = θ p / 3 ( 0 ≤ j ≤ 2 ) , \mathbb{P}(Z_p=0)=1-\theta_p,\qquad\mathbb{P}(Z_p=\omega^j)=\theta_p/3\quad(0\le j\le2), P ( Z p = 0 ) = 1 − θ p , P ( Z p = ω j ) = θ p /3 ( 0 ≤ j ≤ 2 ) , θ p = { 9 / 11 , p = 3 , p / ( p + 2 ) , p ≡ 1 ( m o d 3 ) , 1 , p ≡ 2 ( m o d 3 ) . (138) \theta_p=
\begin{cases}
9/11, & p=3,\\
p/(p+2), & p\equiv1\pmod3,\\
1, & p\equiv2\pmod3.
\end{cases} \tag*{(138)} θ p = ⎩ ⎨ ⎧ 9/11 , p / ( p + 2 ) , 1 , p = 3 , p ≡ 1 ( mod 3 ) , p ≡ 2 ( mod 3 ) . (138) Uniformly for every positive integer n n n
1 N ( Y ) ∑ χ ∈ C ( Y ) χ ( n ) = 1 n is a cube ∏ p ∣ n θ p + O ( Y − 1 / 4 n ) . (139) \frac{1}{N(Y)}\sum_{\chi\in\mathcal{C}(Y)}\chi(n)=1_{n\text{ is a cube}}\prod_{p\mid n}\theta_p+O(Y^{-1/4}n). \tag*{(139)} N ( Y ) 1 χ ∈ C ( Y ) ∑ χ ( n ) = 1 n is a cube p ∣ n ∏ θ p + O ( Y − 1/4 n ) . (139) The same assertion holds for Y < q χ ≤ 2 Y Y<q_\chi\le2Y Y < q χ ≤ 2 Y χ ( p j ) \chi(p^j) χ ( p j ) χ ( p j ) ‾ \overline{\chi(p^j)} χ ( p j ) n ( W ) n(W) n ( W ) O ( Y − 1 / 4 n ( W ) 2 ) O(Y^{-1/4}n(W)^2) O ( Y − 1/4 n ( W ) 2 )
Proof. We give the noncube estimate directly, since a splitting-type count alone does not determine oriented character values. Let O = Z [ ω ] \mathcal{O}=\mathbb{Z}[\omega] O = Z [ ω ] [4 ] , Lemma 2.1 and §2.1, characters in C 0 ( Y ) \mathcal{C}_0(Y) C 0 ( Y ) c ≡ 1 ( m o d 3 ) c\equiv1\pmod3 c ≡ 1 ( mod 3 ) N ( c ) ≤ Y N(c) \le Y N ( c ) ≤ Y ( ⋅ / c ) 3 (\cdot/c)_3 ( ⋅ / c ) 3
ψ n ( ( c ) ) = ( n / c ) 3 ( c ≡ 1 ( m o d 3 ) ) \psi_n((c)) = (n/c)_3 \qquad(c \equiv1 \pmod{3}) ψ n (( c )) = ( n / c ) 3 ( c ≡ 1 ( mod 3 )) of modulus 9 n O 9n\mathcal{O} 9 n O n n n Q ( ω ) \mathbb{Q}(\omega) Q ( ω ) p ≠ 3 p \ne3 p = 3 v p ( n ) v_p(n) v p ( n ) 2 v 3 ( n ) 2v_3(n) 2 v 3 ( n )
We retain the original imprimitive support: the value is zero whenever ( c , n ) ≠ 1 (c,n) \ne1 ( c , n ) = 1 R = 9 n R=9n R = 9 n f ( c ) = 1 c ≡ 1 ( 3 ) ( n / c ) 3 f(c)=\mathbf{1}_{c\equiv1(3)}(n/c)_3 f ( c ) = 1 c ≡ 1 ( 3 ) ( n / c ) 3 R O R\mathcal{O} R O O \mathcal{O} O U = ( O / R O ) × U=(\mathcal{O}/R\mathcal{O})^\times U = ( O / R O ) × H = ker ( U → ( O / 3 O ) × ) H=\ker(U\to(\mathcal{O}/3\mathcal{O})^\times) H = ker ( U → ( O /3 O ) × ) U U U H H H U U U H H H 9 n 9n 9 n
Tiling the disc N ( c ) ≤ V N(c)\le V N ( c ) ≤ V R O R\mathcal{O} R O O ( V / R + 1 ) O(\sqrt{V}/R+1) O ( V / R + 1 ) O ( R 2 ) O(R^2) O ( R 2 ) V ≥ R 2 V\ge R^2 V ≥ R 2 O ( n V ) O(n\sqrt{V}) O ( n V ) V < R 2 V<R^2 V < R 2 O ( V ) O(V) O ( V ) V ≥ 1 V\ge1 V ≥ 1
B n ( V ) : = ∑ c ≡ 1 ( 3 ) N ( c ) ≤ V ( n / c ) 3 ≪ min ( V , n V ) , n not a cube . (140) B_n(V):=\sum_{\substack{c\equiv1(3)\\N(c)\le V}}(n/c)_3\ll\min(V,n\sqrt{V}),\qquad n\text{ not a cube}. \tag*{(140)} B n ( V ) := c ≡ 1 ( 3 ) N ( c ) ≤ V ∑ ( n / c ) 3 ≪ min ( V , n V ) , n not a cube . (140) We spell out how the restrictions on c c c d ≡ 1 ( m o d 3 ) d\equiv1\pmod{3} d ≡ 1 ( mod 3 ) ∣ d ∣ |d| ∣ d ∣ c = d b c=db c = d b ( d , b ) = 1 (d,b)=1 ( d , b ) = 1 b b b b = l b 1 b=\mathfrak{l}b_1 b = l b 1 ( b 1 , d ) = 1 (b_1,d)=1 ( b 1 , d ) = 1 e ∣ ( d ) \mathfrak{e}\mid(d) e ∣ ( d ) n n n d d d n n n ( n / d ) 3 = 1 (n/d)_3=1 ( n / d ) 3 = 1
∣ ∑ χ ∈ C 0 ( Y ) χ ( n ) ∣ ≪ 1 + ∑ d ≡ 1 ( 3 ) ∣ d ∣ ≤ Y μ 2 ( ∣ d ∣ ) = 1 ∑ e ∣ ( d ) ∑ l min ( Y d 2 N ( e ) ( N l ) 2 , n Y ∣ d ∣ N e N l ) . (141) \left|\sum_{\chi\in\mathcal{C}_0(Y)}\chi(n)\right|
\ll1+\sum_{\substack{d\equiv1(3)\\ |d|\le\sqrt{Y}\\ \mu^2(|d|)=1}}\sum_{\mathfrak{e}\mid(d)}\sum_{\mathfrak{l}}\min\left(\frac{Y}{d^2N(\mathfrak{e})(N\mathfrak{l})^2},\frac{n\sqrt{Y}}{|d|\sqrt{N\mathfrak{e}N\mathfrak{l}}}\right). \tag*{(141)} χ ∈ C 0 ( Y ) ∑ χ ( n ) ≪ 1 + d ≡ 1 ( 3 ) ∣ d ∣ ≤ Y μ 2 ( ∣ d ∣ ) = 1 ∑ e ∣ ( d ) ∑ l ∑ min ( d 2 N ( e ) ( N l ) 2 Y , ∣ d ∣ N e N l n Y ) . (141) Only ideals prime to 3 need occur, but dropping this restriction is harmless. Terms whose implied norm bound is below one are zero.
The number of ideals of norm at most U U U O ( U ) O(U) O ( U ) y = Y / ( ∣ d ∣ N e ) y=\sqrt{Y}/(|d|\sqrt{N\mathfrak{e}}) y = Y / ( ∣ d ∣ N e ) N l N\mathfrak{l} N l y / n y/n y / n
∑ l min { y 2 / ( N l ) 2 , n y / N l } ≪ n y log ( 2 Y ) . \sum_{\mathfrak{l}}\min\{y^2/(N\mathfrak{l})^2,ny/N\mathfrak{l}\}\ll ny\log(2Y). l ∑ min { y 2 / ( N l ) 2 , n y / N l } ≪ n y log ( 2 Y ) . When y < n y<n y < n y 2 / ( N l ) 2 y^2/(N\mathfrak{l})^2 y 2 / ( N l ) 2 O ( y 2 ) ≤ O ( n y ) O(y^2)\le O(ny) O ( y 2 ) ≤ O ( n y )
∑ d ≤ D μ 2 ( d ) = 1 1 d ∑ e ∣ ( d ) ( N e ) − 1 / 2 ≪ log ( 2 D ) . (142) \sum_{\substack{d\le D\\ \mu^2(d)=1}}\frac{1}{d}\sum_{\mathfrak{e}\mid(d)}(N\mathfrak{e})^{-1/2}\ll\log(2D). \tag*{(142)} d ≤ D μ 2 ( d ) = 1 ∑ d 1 e ∣ ( d ) ∑ ( N e ) − 1/2 ≪ log ( 2 D ) . (142) Indeed the inner multiplicative function has local value ( 1 + p − 1 / 2 ) 2 (1+p^{-1/2})^2 ( 1 + p − 1/2 ) 2 1 + p − 1 1+p^{-1} 1 + p − 1 1 ∗ h 1*h 1 ∗ h ∏ p ( 1 + h ( p ) / p ) \prod_p(1+h(p)/p) ∏ p ( 1 + h ( p ) / p ) h h h (142) . Thus
S 0 ( Y ; n ) : = ∑ χ ∈ C 0 ( Y ) χ ( n ) ≪ n Y log 2 ( 2 Y ) ( n not a cube ) . (143) S_0(Y;n):=\sum_{\chi\in\mathcal{C}_0(Y)}\chi(n)\ll n\sqrt{Y}\log^2(2Y)\qquad(n\text{ not a cube}). \tag*{(143)} S 0 ( Y ; n ) := χ ∈ C 0 ( Y ) ∑ χ ( n ) ≪ n Y log 2 ( 2 Y ) ( n not a cube ) . (143) For cubes we use only the unconditional counting assertion of [11 Proposition 2.1] , specialized to its family of conductors prime to 3:
N 0 ( Y ) = K 0 Y + O δ ( Y 5 / 8 + δ ) , N_0(Y)=K_0Y+O_\delta(Y^{5/8+\delta}), N 0 ( Y ) = K 0 Y + O δ ( Y 5/8 + δ ) , S 0 ( Y ; n ) = K 0 Y ∏ p ∣ n p ≡ 1 ( 3 ) p p + 2 + O δ ( 3 ω ( n ) Y 5 / 8 + δ ) ( n a cube ) , (144) S_0(Y;n)=K_0Y\prod_{\substack{p\mid n\\p\equiv1(3)}}\frac{p}{p+2}+O_\delta(3^{\omega(n)}Y^{5/8+\delta})\qquad(n\text{ a cube}), \tag*{(144)} S 0 ( Y ; n ) = K 0 Y p ∣ n p ≡ 1 ( 3 ) ∏ p + 2 p + O δ ( 3 ω ( n ) Y 5/8 + δ ) ( n a cube ) , (144) where
K 0 = 2 3 L ( 1 , χ − 3 ) ∏ p ≡ 1 ( 3 ) ( 1 − 3 p − 2 + 2 p − 3 ) ∏ p ≡ 2 ( 3 ) ( 1 − p − 2 ) > 0. K_0=\frac{2}{3}L(1,\chi_{-3})\prod_{p\equiv1(3)}(1-3p^{-2}+2p^{-3})\prod_{p\equiv2(3)}(1-p^{-2})>0. K 0 = 3 2 L ( 1 , χ − 3 ) p ≡ 1 ( 3 ) ∏ ( 1 − 3 p − 2 + 2 p − 3 ) p ≡ 2 ( 3 ) ∏ ( 1 − p − 2 ) > 0. The constant is the residue at one of ∏ p ≡ 1 ( 3 ) ( 1 + 2 p − s ) \prod_{p\equiv1(3)}(1+2p^{-s}) ∏ p ≡ 1 ( 3 ) ( 1 + 2 p − s ) ( q , n ) = 1 (q,n)=1 ( q , n ) = 1
It remains to include the exact 3-part of the conductor. Let ξ , ξ ˉ \xi,\bar{\xi} ξ , ξ ˉ p ≡ 1 ( m o d 3 ) p\equiv1\pmod{3} p ≡ 1 ( mod 3 ) b ( Y ) = 1 Y ≥ 9 b(Y)=1_{Y\ge9} b ( Y ) = 1 Y ≥ 9
S ( Y ; n ) = S 0 ( Y ; n ) + [ ξ ( n ) + ξ ˉ ( n ) ] { S 0 ( Y / 9 ; n ) + b ( Y ) } , S(Y;n)=S_0(Y;n)+[\xi(n)+\bar{\xi}(n)]\{S_0(Y/9;n)+b(Y)\}, S ( Y ; n ) = S 0 ( Y ; n ) + [ ξ ( n ) + ξ ˉ ( n )] { S 0 ( Y /9 ; n ) + b ( Y )} , N ( Y ) = N 0 ( Y ) + 2 N 0 ( Y / 9 ) + 2 b ( Y ) . (145) N(Y)=N_0(Y)+2N_0(Y/9)+2b(Y). \tag*{(145)} N ( Y ) = N 0 ( Y ) + 2 N 0 ( Y /9 ) + 2 b ( Y ) . (145) In particular N ( Y ) = K Y + O δ ( Y 5 / 8 + δ ) N(Y)=KY+O_\delta(Y^{5/8+\delta}) N ( Y ) = K Y + O δ ( Y 5/8 + δ ) K = 11 K 0 / 9 K=11K_0/9 K = 11 K 0 /9 n n n 21 3 ∤ n 2 1_{3\nmid n} 2 1 3 ∤ n 3 ∤ n 3\nmid n 3 ∤ n 9 / 11 9/11 9/11 3 ∣ n 3\mid n 3 ∣ n n n n (143) and (145) give the same noncube bound for S ( Y ; n ) S(Y;n) S ( Y ; n ) δ = 1 / 16 \delta=1/16 δ = 1/16 3 ω ( n ) ≤ n 3^{\omega(n)}\le n 3 ω ( n ) ≤ n Y − 1 / 2 log 2 ( 2 Y ) ≪ Y − 1 / 4 Y^{-1/2}\log^2(2Y)\ll Y^{-1/4} Y − 1/2 log 2 ( 2 Y ) ≪ Y − 1/4 N ( Y ) N(Y) N ( Y ) (139) with one absolute constant for all n n n
Finally, z ˉ = z 2 \bar{z}=z^2 z ˉ = z 2 z ∈ { 0 , 1 , ω , ω 2 } z\in\{0,1,\omega,\omega^2\} z ∈ { 0 , 1 , ω , ω 2 } z = 0 z=0 z = 0 χ ( n W ) \chi(n_W) χ ( n W ) n W ≤ n ( W ) 2 n_W\le n(W)^2 n W ≤ n ( W ) 2 b Y bY bY a Y aY aY a Y < q χ ≤ b Y aY<q_\chi\le bY aY < q χ ≤ bY 0 < a < b 0<a<b 0 < a < b a , b a,b a , b q χ 2 q_\chi^2 q χ 2 2 \sqrt{2} 2
For completeness, this lemma also determines joint phases, rather than only joint splitting types. On the four-point support,
1 z = 0 = 1 − z 3 , 1 z = ζ = 1 3 ( ζ 3 + ζ ˉ z + ζ ˉ 2 z 2 ) ( ζ 3 = 1 ) . \mathbf{1}_{z=0}=1-z^3,\qquad\mathbf{1}_{z=\zeta}=\frac{1}{3}(\zeta^3+\bar{\zeta}z+\bar{\zeta}^2z^2)\qquad(\zeta^3=1). 1 z = 0 = 1 − z 3 , 1 z = ζ = 3 1 ( ζ 3 + ζ ˉ z + ζ ˉ 2 z 2 ) ( ζ 3 = 1 ) . Consequently a phase specification on S S S O ( 2 ∣ S ∣ ( ∏ p ∈ S p ) 3 Y − 1 / 4 ) O(2^{|S|}(\prod_{p\in S}p)^{3}Y^{-1/4}) O ( 2 ∣ S ∣ ( ∏ p ∈ S p ) 3 Y − 1/4 )
Real cyclic cubic augmentation
Each cyclic cubic field K K K χ , χ ‾ \chi,\overline{\chi} χ , χ q K q_K q K
D K = q K 2 , Z K ( s ) : = ζ K ( s ) ζ ( s ) = L ( s , χ ) L ( s , χ ‾ ) , Λ K ( s ) = q K s Γ R ( s ) 2 Z K ( s ) = Λ K ( 1 − s ) . (146) D_K=q_K^2,\qquad Z_K(s):=\frac{\zeta_K(s)}{\zeta(s)}=L(s,\chi)L(s,\overline{\chi}),\qquad\Lambda_K(s)=q_K^s\Gamma_{\mathbb{R}}(s)^2Z_K(s)=\Lambda_K(1-s). \tag*{(146)} D K = q K 2 , Z K ( s ) := ζ ( s ) ζ K ( s ) = L ( s , χ ) L ( s , χ ) , Λ K ( s ) = q K s Γ R ( s ) 2 Z K ( s ) = Λ K ( 1 − s ) . (146) Every cubic character is even. Thus Z K Z_K Z K Z K ( x ) = ∣ L ( x , χ ) ∣ 2 Z_K(x)=\lvert L(x,\chi)\rvert^2 Z K ( x ) = ∣ L ( x , χ ) ∣ 2 x x x Q K = q K 2 Q_K=q_K^2 Q K = q K 2 L L L Z K Z_K Z K
Proposition 6.7. The uniform family of cyclic cubic fields with Y < q K ≤ 2 Y Y<q_K\le2Y Y < q K ≤ 2 Y (120) with ( κ , A , C 0 ) = ( 1 / 4 , 2 , 2 ) (\kappa,A,C_0)=(1/4,2,2) ( κ , A , C 0 ) = ( 1/4 , 2 , 2 ) g / 2 g/\sqrt{2} g / 2 Y Y Y ( 11 K 0 / 18 ) Y + O ( Y 11 / 16 ) (11K_0/18)Y+O(Y^{11/16}) ( 11 K 0 /18 ) Y + O ( Y 11/16 )
Proof. The local coefficients are A K ( p j ) = χ ( p ) j + χ ‾ ( p ) j A_K(p^j)=\chi(p)^j+\overline{\chi}(p)^j A K ( p j ) = χ ( p ) j + χ ( p ) j r r r 2 r 2^r 2 r
At a prime, the split, inert, and ramified states have respective probabilities θ p / 3 \theta_p/3 θ p /3 2 θ p / 3 2\theta_p/3 2 θ p /3 1 − θ p 1-\theta_p 1 − θ p 3 ∤ j 3\nmid j 3 ∤ j 2 , − 1 , 0 2,-1,0 2 , − 1 , 0 3 ∣ j 3\mid j 3 ∣ j 2 , 2 , 0 2,2,0 2 , 2 , 0
E A ( p ) = E A ( p 2 ) = 0 , E A ( p ) 2 = 2 θ p = 2 + O ( p − 1 ) . (147) \mathbb{E}A(p)=\mathbb{E}A(p^2)=0,\qquad\mathbb{E}A(p)^2=2\theta_p=2+O(p^{-1}). \tag*{(147)} E A ( p ) = E A ( p 2 ) = 0 , E A ( p ) 2 = 2 θ p = 2 + O ( p − 1 ) . (147) The mean starts with j = 3 j=3 j = 3 2 a b / ( a + b ) 2 + O ( a b ) 2ab/(a+b)^2+O(ab) 2 ab / ( a + b ) 2 + O ( ab )
3 a ( log p ) u ( 1 + u ) 1 − u 3 , u = p − 1 / 2 − a , 3a(\log p)\frac{u(1+u)}{1-u^3},\qquad u=p^{-1/2-a}, 3 a ( log p ) 1 − u 3 u ( 1 + u ) , u = p − 1/2 − a , which is comparable to a ( log p ) u a(\log p)u a ( log p ) u p = 3 p=3 p = 3 n ( W ) 2 n(W)^2 n ( W ) 2 e C ( ℓ + 1 ) e^{C(\ell+1)} e C ( ℓ + 1 ) m ℓ = o ( log Y ) m\ell=o(\log Y) m ℓ = o ( log Y )
Lemma 6.8 (Uniform analytic specifications). Every primitive function used above has an entire completion of order one. Its gamma parameters belong to a fixed finite set, and on every fixed strip − B ≤ ℜ s ≤ B -B\le\Re s\le B − B ≤ ℜ s ≤ B
∣ L F ( s ) ∣ ≤ C B [ Q F ( 2 + ∣ ℑ s ∣ ) ] C B . (148) \lvert L_F(s)\rvert\le C_B[Q_F(2+\lvert\Im s\rvert)]^{C_B}. \tag*{(148)} ∣ L F ( s )∣ ≤ C B [ Q F ( 2 + ∣ ℑ s ∣) ] C B . (148) The same assertions, with a contragredient functional equation, hold for each individual primitive cubic character. These statements do not require GRH.
Proof. For the modular and S 3 S_3 S 3 L L L [23 §2.14.5] ; we only use its individual order statement, not a claim of uniformity in the representation for its displayed strip estimate. For a primitive nonprincipal Dirichlet character, the theta transformation and Mellin integral prove entireness and (150) ; see [19 Chapter 11] . Splitting that integral at one and transforming the lower half expresses the completion as two integrals over [ 1 , ∞ ) [1,\infty) [ 1 , ∞ ) q q q C q e − c q y 2 C_q e^{-c_q y^2} C q e − c q y 2 ∣ s ∣ ≤ R |s| \le R ∣ s ∣ ≤ R
C q ∫ 1 ∞ e − c q y 2 y R + C d y ≤ exp ( C q R log ( R + 2 ) ) . C_q \int_1^\infty e^{-c_q y^2}y^{R+C}\,dy \le\exp(C_q R\log(R+2)). C q ∫ 1 ∞ e − c q y 2 y R + C d y ≤ exp ( C q R log ( R + 2 )) . This proves order at most one directly; division by the gamma factor preserves finite order. It also proves finite order for the real product of two conjugates.
We obtain the required uniformity independently. At a fixed right boundary b > max ( B , 2 ) b>\max(B,2) b > max ( B , 2 ) ∣ L F ( b + i t ) ∣ ≤ ζ ( b ) d |L_F(b+it)|\le\zeta(b)^d ∣ L F ( b + i t ) ∣ ≤ ζ ( b ) d a < min ( − B , − 1 ) a<\min(-B,-1) a < min ( − B , − 1 )
∣ L F ( a + i t ) ∣ ≤ C Q F 1 / 2 − a ( 2 + ∣ t ∣ ) C . |L_F(a+it)| \le C Q_F^{1/2-a}(2+|t|)^C. ∣ L F ( a + i t ) ∣ ≤ C Q F 1/2 − a ( 2 + ∣ t ∣ ) C . The contragredient has the same right-boundary Euler bound. Choose a fixed polynomial ( s + c ) J (s+c)^J ( s + c ) J L F ( s ) / ( s + c ) J L_F(s)/(s+c)^J L F ( s ) / ( s + c ) J (148) . The individual constants in the initial finite-order bound do not survive this argument.
Finally Stirling’s formula and the Euler product bound the completion in the right half-plane by exp ( C F R log ( R + 2 ) ) \exp(C_F R\log(R+2)) exp ( C F R log ( R + 2 )) ∣ s ∣ ≤ R |s|\le R ∣ s ∣ ≤ R (148) in the intervening fixed strip. The finitely many apparent gamma poles there are removable and are filled by the maximum principle. The completion thus has order at most one. Along the positive real axis L F ( σ ) → 1 L_F(\sigma)\to1 L F ( σ ) → 1
log ∣ Λ F ( σ ) ∣ = ( d / 2 ) σ log σ + O F ( σ ) , \log|\Lambda_F(\sigma)|=(d/2)\sigma\log\sigma+O_F(\sigma), log ∣ Λ F ( σ ) ∣ = ( d /2 ) σ log σ + O F ( σ ) , so the order is exactly one. For modular twists the gamma shifts are ( k − 1 ) / 2 , ( k + 1 ) / 2 (k-1)/2,(k+1)/2 ( k − 1 ) /2 , ( k + 1 ) /2 S 3 S_3 S 3 ( 0 , 0 ) (0,0) ( 0 , 0 ) ( 0 , 1 ) (0,1) ( 0 , 1 ) ( 0 , 0 ) (0,0) ( 0 , 0 )
Corollary 6.9 (The real-family constants). Assume GRH for the primitive L L L Proposition 6.3 . In particular, outside a subset of density zero, each of the distinct-zero count, the count with multiplicity on [ 1 / 2 , 1 ] [1/2,1] [ 1/2 , 1 ] ( 1 / 2 , 1 ) (1/2,1) ( 1/2 , 1 )
( c + o ( 1 ) ) log log Q F , c = { e − 1 / 2 / ( π 2 ) , modular twists and non-Galois cubic fields , 1 / ( π 2 ) , real cyclic cubic augmentation . (c+o(1))\log\log Q_F,\qquad
c=
\begin{cases}
e^{-1/2}/(\pi\sqrt{2}), & \text{modular twists and non-Galois cubic fields},\\
1/(\pi\sqrt{2}), & \text{real cyclic cubic augmentation}.
\end{cases} ( c + o ( 1 )) log log Q F , c = { e − 1/2 / ( π 2 ) , 1/ ( π 2 ) , modular twists and non-Galois cubic fields , real cyclic cubic augmentation . The same conclusion holds in cumulative conductor or discriminant families, with their original counting measure. More precisely, on each dyadic family, put H = log log Y H=\log\log Y H = log log Y H F = log log Q F H_F=\log\log Q_F H F = log log Q F 0 < ε < 1 / 16 0<\varepsilon<1/16 0 < ε < 1/16 H − 1 / 2 + ε ≤ z ≤ 1 H^{-1/2+\varepsilon}\le z\le1 H − 1/2 + ε ≤ z ≤ 1
P { max C ∈ { R d i s t , R m u l t , S } ∣ C − c H F ∣ > A ε z H F } ≤ C ε log H H z 2 . (149) \mathbb{P}\left\{\max_{C\in\{R^{\mathrm{dist}},R^{\mathrm{mult}},S\}} |C-cH_F|>A_\varepsilon zH_F\right\}
\le C_\varepsilon\frac{\log H}{H z^2}. \tag*{(149)} P { C ∈ { R dist , R mult , S } max ∣ C − c H F ∣ > A ε z H F } ≤ C ε H z 2 log H . (149) The constants may depend on the fixed modular form or the chosen discriminant sign, but not on z z z
Proof. The arithmetic word and polynomial estimates, local Gaussian limits, uniform density inputs, and fixed-degree analytic specifications have all been verified above. The common Selberg and central estimates under GRH therefore apply. The Rice intensity for a process m / 2 + v g / 2 m/2+\sqrt{v}\,g/2 m /2 + v g /2 e − m 2 / ( 2 v ) / ( π 2 ) e^{-m^2/(2v)}/(\pi\sqrt{2}) e − m 2 / ( 2 v ) / ( π 2 ) ( m , v ) = ( − 1 , 1 ) , ( 1 , 1 ) , ( 0 , 2 ) (m,v)=(-1,1),(1,1),(0,2) ( m , v ) = ( − 1 , 1 ) , ( 1 , 1 ) , ( 0 , 2 ) log log D K = log log q K + log 2 \log\log D_K = \log\log q_K + \log2 log log D K = log log q K + log 2 (131) gives the same bounded change. These changes do not affect the leading term. Cumulative families follow by discarding the parameters below Y \sqrt{Y} Y L L L S 3 S_3 S 3 2 2 2 − 1 -1 − 1
Real zeros for individual complex cubic characters
The real augmentation L ( s , χ ) L ( s , χ ‾ ) L(s,\chi)L(s,\overline{\chi}) L ( s , χ ) L ( s , χ ) L ( s , χ ) L(s,\chi) L ( s , χ ) L L L
Let C X = { χ : X / 2 < q χ ≤ X , χ primitive of exact order 3 } C_X=\{\chi:X/2<q_\chi\le X,\ \chi\text{ primitive of exact order }3\} C X = { χ : X /2 < q χ ≤ X , χ primitive of exact order 3 } L = log X L=\log X L = log X H = log log X H=\log\log X H = log log X H χ = log log q χ H_\chi=\log\log q_\chi H χ = log log q χ ∣ C X ∣ |C_X| ∣ C X ∣
Λ ( s , χ ) = q χ s / 2 Γ R ( s ) L ( s , χ ) = ϵ χ Λ ( 1 − s , χ ‾ ) , ∣ ϵ χ ∣ = 1. (150) \Lambda(s,\chi)=q_\chi^{s/2}\Gamma_{\mathbb R}(s)L(s,\chi)=\epsilon_\chi\Lambda(1-s,\overline{\chi}),\qquad|\epsilon_\chi|=1. \tag*{(150)} Λ ( s , χ ) = q χ s /2 Γ R ( s ) L ( s , χ ) = ϵ χ Λ ( 1 − s , χ ) , ∣ ϵ χ ∣ = 1. (150) The functions are entire of order one and have polynomial growth in fixed vertical strips. Cubic characters are even, so there is no parity shift in (150) . Put
L χ ( s ) = − L ′ / L ( s , χ ) , G χ ( w ) = e − w L χ ( 1 / 2 + e − w ) . \mathcal{L}_\chi(s)=-L'/L(s,\chi),\qquad G_\chi(w)=e^{-w}\mathcal{L}_\chi(1/2+e^{-w}). L χ ( s ) = − L ′ / L ( s , χ ) , G χ ( w ) = e − w L χ ( 1/2 + e − w ) . Under GRH, G χ G_\chi G χ ∣ ℑ w ∣ < π / 2 |\Im w|<\pi/2 ∣ℑ w ∣ < π /2
Theorem 7.1 (A sublinear real-zero bound). For every fixed 0 < ϵ < 1 / 16 0<\epsilon<1/16 0 < ϵ < 1/16 A ϵ A_\epsilon A ϵ
P χ ∈ C X { R L ′ ( ⋅ , χ ) m u l t ( [ 1 / 2 , 1 ] ) > A ϵ H χ 1 / 2 + ϵ } ≪ ϵ log H H 1 / 2 + ϵ + H − 2 log H . (151) \mathbb{P}_{\chi\in C_X}\left\{R^{\mathrm{mult}}_{L'(\cdot,\chi)}([1/2,1])>A_\epsilon H_\chi^{1/2+\epsilon}\right\}\ll_\epsilon\frac{\log H}{H^{1/2+\epsilon}}+H^{-2}\log H. \tag*{(151)} P χ ∈ C X { R L ′ ( ⋅ , χ ) mult ([ 1/2 , 1 ]) > A ϵ H χ 1/2 + ϵ } ≪ ϵ H 1/2 + ϵ log H + H − 2 log H . (151) In particular, the distinct and multiple real-zero counts are o ( log log q χ ) o(\log\log q_\chi) o ( log log q χ ) 1 − O ( H − 3 ) 1-O(H^{-3}) 1 − O ( H − 3 ) L ′ ( s , χ ) L'(s,\chi) L ′ ( s , χ )
50 log H ≤ − log ( s − 1 / 2 ) ≤ H − 121 log H . (152) 50\log H\le-\log(s-1/2)\le H-121\log H. \tag*{(152)} 50 log H ≤ − log ( s − 1/2 ) ≤ H − 121 log H . (152) The full-interval estimate also holds in the cumulative family q χ ≤ X q_\chi\le X q χ ≤ X
The two-dimensional random limit
Use the independent variables in (138) and define, in ℜ a > 0 \Re a>0 ℜ a > 0
G E ( a ) = a ∑ p ∑ j ≥ 1 Z p j ( log p ) p − j ( 1 / 2 + a ) . \mathcal{G}_E(a)=a\sum_p\sum_{j\ge1}Z_p^j(\log p)p^{-j(1/2+a)}. G E ( a ) = a p ∑ j ≥ 1 ∑ Z p j ( log p ) p − j ( 1/2 + a ) . The centered prime-linear series converges locally in L 2 L^2 L 2
Lemma 7.2 (Complex Gaussian comparison). For positive real a , b a,b a , b
E G E ( a ) = O ( a ) , \mathbb{E}\mathcal{G}_E(a)=O(a), E G E ( a ) = O ( a ) , E [ ( G E ( a ) − E G E ( a ) ) ( G E ( b ) − E G E ( b ) ) ‾ ] = a b ( a + b ) 2 + O ( a b ) , \mathbb{E}\left[(\mathcal{G}_E(a)-\mathbb{E}\mathcal{G}_E(a))\overline{(\mathcal{G}_E(b)-\mathbb{E}\mathcal{G}_E(b))}\right]=\frac{ab}{(a+b)^2}+O(ab), E [ ( G E ( a ) − E G E ( a )) ( G E ( b ) − E G E ( b )) ] = ( a + b ) 2 ab + O ( ab ) , E [ ( G E ( a ) − E G E ( a ) ) ( G E ( b ) − E G E ( b ) ) ] = O ( a b ) . (153) \mathbb{E}\left[(\mathcal{G}_E(a)-\mathbb{E}\mathcal{G}_E(a))(\mathcal{G}_E(b)-\mathbb{E}\mathcal{G}_E(b))\right]=O(ab). \tag*{(153)} E [ ( G E ( a ) − E G E ( a )) ( G E ( b ) − E G E ( b )) ] = O ( ab ) . (153) In the variable t = − log a t=-\log a t = − log a g 1 / ( 2 2 ) g_1/(2\sqrt{2}) g 1 / ( 2 2 ) g 2 / ( 2 2 ) g_2/(2\sqrt{2}) g 2 / ( 2 2 ) g 1 , g 2 g_1,g_2 g 1 , g 2 g g g 1 8 I 2 \frac{1}{8}I_2 8 1 I 2
For a fixed collection of d d d C 3 C^3 C 3 Φ \Phi Φ M j M_j M j j j j Φ \Phi Φ b ∗ = max a i b_*=\max a_i b ∗ = max a i
C ( d b ∗ M 1 + d 2 b ∗ 2 M 2 + d 3 b ∗ 3 M 3 ) . (154) C\left(db_*M_1+d^2b_*^2M_2+d^3b_*^3M_3\right). \tag*{(154)} C ( d b ∗ M 1 + d 2 b ∗ 2 M 2 + d 3 b ∗ 3 M 3 ) . (154) Normalized analytic suprema and fixed derivatives on fixed sectorial discs have bounded moments of every fixed order.
Proof. The elementary identities are E Z p = E Z p 2 = 0 \mathbb{E}Z_p=\mathbb{E}Z_p^2=0 E Z p = E Z p 2 = 0 E ∣ Z p ∣ 2 = θ p = 1 + O ( p − 1 ) \mathbb{E}|Z_p|^2=\theta_p=1+O(p^{-1}) E ∣ Z p ∣ 2 = θ p = 1 + O ( p − 1 ) j = 3 j=3 j = 3 O ( a ) O(a) O ( a )
a b ∑ p θ p ( log p ) 2 p − 1 − a − b = a b ( a + b ) 2 + O ( a b ) . ab\sum_p\theta_p(\log p)^2p^{-1-a-b}=\frac{ab}{(a+b)^2}+O(ab). ab p ∑ θ p ( log p ) 2 p − 1 − a − b = ( a + b ) 2 ab + O ( ab ) . The error from θ p − 1 \theta_p-1 θ p − 1 E Z p 2 = 0 \mathbb{E}Z_p^2=0 E Z p 2 = 0 (153) . The real and imaginary covariance matrices now give the stated limit; their cross covariance tends to zero.
For the smooth comparison, regard all the coordinates from a given prime as a single real vector. Subtract its mean and replace this whole vector by a Gaussian with the same covariance. A third-order Taylor expansion, with its integral remainder, gives an error at most C d 3 M 3 Cd^3M_3 C d 3 M 3 O ( b ∗ 3 ) O(b_*^3) O ( b ∗ 3 ) ∑ p ( log p ) 3 p − 3 / 2 < ∞ \sum_p(\log p)^3p^{-3/2}<\infty ∑ p ( log p ) 3 p − 3/2 < ∞ C d b ∗ M 1 Cdb_*M_1 C d b ∗ M 1 O ( b ∗ 2 ) O(b_*^2) O ( b ∗ 2 ) C d 2 b ∗ 2 M 2 Cd^2b_*^2M_2 C d 2 b ∗ 2 M 2 L 2 L^2 L 2 (154) . The argument never assumes independence of real and imaginary parts at the same prime.
Each real projection of the centered prime sum is a sum of independent bounded variables whose squared ranges have bounded sum after normalization. The exponential-moment inequality gives the required fixed-order moments. Apply Cauchy’s formula on a slightly larger disc, then Minkowski’s inequality, to obtain analytic supremum and derivative moments. □
Lemma 7.3 (Finite characteristic-function comparison). Let P χ \mathbf{P}_{\chi} P χ p j ≤ e 3 ℓ p^j\le e^{3\ell} p j ≤ e 3 ℓ P E \mathbf{P}_E P E ∥ τ ∥ 1 ≤ T \|\tau\|_1\le T ∥ τ ∥ 1 ≤ T M ≥ 1 M\ge1 M ≥ 1
∣ E e i ⟨ τ , P χ ⟩ − E e i ⟨ τ , P E ⟩ ∣ ≤ 2 ( C T M ) 2 M + C X − 1 / 4 exp { C M ( ℓ + log ( 2 + T ) ) } . (155) \left|\mathbb{E}e^{i\langle\tau,\mathbf{P}_{\chi}\rangle}-\mathbb{E}e^{i\langle\tau,\mathbf{P}_E\rangle}\right|
\le2\left(\frac{CT}{\sqrt{M}}\right)^{2M}+CX^{-1/4}\exp\{CM(\ell+\log(2+T))\}. \tag*{(155)} E e i ⟨ τ , P χ ⟩ − E e i ⟨ τ , P E ⟩ ≤ 2 ( M C T ) 2 M + C X − 1/4 exp { C M ( ℓ + log ( 2 + T ))} . (155) The same comparison gives normalized polynomial moments at every fixed complex location in a fixed sector. At the auxiliary height anchor a 0 = 4 / ℓ a_0=4/\ell a 0 = 4/ ℓ
∥ a P χ ( 1 / 2 + a ) ∥ q ≤ C q , ∥ a 0 P χ ( 1 / 2 + a 0 + i t ) ∥ q ≤ C q (156) \|aP_{\chi}(1/2+a)\|_q\le C_q,\qquad\|a_0P_{\chi}(1/2+a_0+it)\|_q\le C_q \tag*{(156)} ∥ a P χ ( 1/2 + a ) ∥ q ≤ C q , ∥ a 0 P χ ( 1/2 + a 0 + i t ) ∥ q ≤ C q (156) for fixed q q q ∣ t ∣ ≤ 1 |t|\le1 ∣ t ∣ ≤ 1 Z χ = L ( ⋅ , χ ) L ( ⋅ , χ ‾ ) Z_{\chi}=L(\cdot,\chi)L(\cdot,\overline{\chi}) Z χ = L ( ⋅ , χ ) L ( ⋅ , χ )
Proof. Expand each real or imaginary coordinate as a linear combination of χ ( p j ) \chi(p^j) χ ( p j ) n ( W ) 2 n(W)^2 n ( W ) 2 e O ( ℓ + 1 ) e^{O(\ell+1)} e O ( ℓ + 1 ) V = ⟨ τ , P ⟩ V=\langle\tau,\mathbf{P}\rangle V = ⟨ τ , P ⟩ 2 M 2M 2 M ( C T M ) 2 M (CT\sqrt{M})^{2M} ( C T M ) 2 M e i V e^{iV} e iV 2 M − 1 2M-1 2 M − 1 ∣ V ∣ 2 M / ( 2 M ) ! |V|^{2M}/(2M)! ∣ V ∣ 2 M / ( 2 M )! ( 2 M ) ! ≥ ( 2 M / e ) 2 M (2M)!\ge(2M/e)^{2M} ( 2 M )! ≥ ( 2 M / e ) 2 M (155) . No growth bound for an exponential of a complex projection is used. The same expansion for fixed even moments proves the first part of (156) . For the second, apply the trace-word estimate in Proposition 6.7. The factors p − i j t p^{-ijt} p − ij t t t t X − 1 / 4 e C q ℓ = o ( 1 ) X^{-1/4}e^{C_q\ell}=o(1) X − 1/4 e C q ℓ = o ( 1 )
Approximation of the actual complex function
Lemma 7.4 (Complex approximation from a real product). Put ℓ = C y L / H 120 \ell=C_yL/H^{120} ℓ = C y L / H 120 C y C_y C y [ 50 log H , H − 121 log H ] [50\log H,H-121\log H] [ 50 log H , H − 121 log H ] w w w G χ ( w ) G_\chi(w) G χ ( w ) a = e − w a=e^{-w} a = e − w
∥ a ( L χ − P χ ) ( 1 / 2 + a ) ∥ q ≤ C q H 121 e − c C y H . (157) \lVert a(\mathcal{L}_\chi-P_\chi)(1/2+a)\rVert_q\le C_qH^{121}e^{-cC_yH}. \tag*{(157)} ∥ a ( L χ − P χ ) ( 1/2 + a ) ∥ q ≤ C q H 121 e − c C y H . (157) Proof. The contour proof of Lemma 3.1 applies to the individual primitive function in (150) . The contour identity itself uses entireness, order one, the functional equation with the contragredient, and fixed-strip growth; it does not require that this individual function be real on the real axis. In the notation of that theorem,
L χ ( s ) = P χ ( s ) + ∑ ρ o f L ( ⋅ , χ ) K ℓ ( ρ − s ) + E t r , χ ( s ) , K ℓ ( z ) = e ℓ z − 2 e 2 ℓ z + e 3 ℓ z ℓ 2 z 3 . (158) \mathcal{L}_\chi(s)=P_\chi(s)+\sum_{\rho\ {\rm of}\ L(\cdot,\chi)}K_\ell(\rho-s)+E_{\mathrm{tr},\chi}(s),\qquad K_\ell(z)=\frac{e^{\ell z}-2e^{2\ell z}+e^{3\ell z}}{\ell^2z^3}. \tag*{(158)} L χ ( s ) = P χ ( s ) + ρ of L ( ⋅ , χ ) ∑ K ℓ ( ρ − s ) + E tr , χ ( s ) , K ℓ ( z ) = ℓ 2 z 3 e ℓ z − 2 e 2 ℓ z + e 3 ℓ z . (158) The trivial-zero progression is that of an even primitive Dirichlet character, and E t r , χ ( s ) ≪ ℓ − 2 e − ℓ ℜ s E_{\mathrm{tr},\chi}(s)\ll\ell^{-2}e^{-\ell\Re s} E tr , χ ( s ) ≪ ℓ − 2 e − ℓ ℜ s Z χ Z_\chi Z χ L ( ⋅ , χ ) L(\cdot,\chi) L ( ⋅ , χ ) 2 + i t 2+it 2 + i t σ + i t \sigma+it σ + i t Z χ Z_\chi Z χ 1 / log ( q χ ( ∣ t ∣ + 2 ) ) 1/\log(q_\chi(|t|+2)) 1/ log ( q χ ( ∣ t ∣ + 2 )) O ( log 2 ( q χ ( ∣ t ∣ + 2 ) ) ) O(\log^2(q_\chi(|t|+2))) O ( log 2 ( q χ ( ∣ t ∣ + 2 ))) ∣ t ∣ − 3 |t|^{-3} ∣ t ∣ − 3 (150) reduces the logarithmic derivative to the convergent Euler derivative for χ ‾ \overline{\chi} χ (158) without a pairing convention for the individual zeros.
Write s = 1 / 2 + a + i t s=1/2+a+it s = 1/2 + a + i t a a a a ≥ a 0 = 4 / ℓ a\ge a_0=4/\ell a ≥ a 0 = 4/ ℓ
∑ ρ o f L ( ⋅ , χ ) ∣ K ℓ ( ρ − s ) ∣ ≤ 4 e − a ℓ ℓ 2 a a 0 ∑ ρ o f Z χ a 0 a 0 2 + ( t − ℑ ρ ) 2 . (159) \sum_{\rho\ {\rm of}\ L(\cdot,\chi)}|K_\ell(\rho-s)|\le\frac{4e^{-a\ell}}{\ell^2aa_0}\sum_{\rho\ {\rm of}\ Z_\chi}\frac{a_0}{a_0^2+(t-\Im\rho)^2}. \tag*{(159)} ρ of L ( ⋅ , χ ) ∑ ∣ K ℓ ( ρ − s ) ∣ ≤ ℓ 2 a a 0 4 e − a ℓ ρ of Z χ ∑ a 0 2 + ( t − ℑ ρ ) 2 a 0 . (159) Indeed ∣ a + i u ∣ − 3 ≤ [ a ( a 2 + u 2 ) ] − 1 |a+iu|^{-3}\le[a(a^2+u^2)]^{-1} ∣ a + i u ∣ − 3 ≤ [ a ( a 2 + u 2 ) ] − 1 a ≥ a 0 a\ge a_0 a ≥ a 0 Z χ Z_\chi Z χ Lemma 3.1 bound it by C ( L + ∣ P Z χ ( 1 / 2 + a 0 + i t ) ∣ ) C(L+|P_{Z_\chi}(1/2+a_0+it)|) C ( L + ∣ P Z χ ( 1/2 + a 0 + i t ) ∣ ) 4 / ( ℓ 2 a a 0 ) = 1 / ( a ℓ ) ≤ 1 / 4 4/(\ell^2aa_0)=1/(a\ell)\le1/4 4/ ( ℓ 2 a a 0 ) = 1/ ( a ℓ ) ≤ 1/4
∣ L χ ( 1 / 2 + a + i t ) − P χ ( 1 / 2 + a + i t ) ∣ ≤ C e − a ℓ { L + ∣ P Z χ ( 1 / 2 + a 0 + i t ) ∣ } . (160) |\mathcal{L}_\chi(1/2+a+it)-P_\chi(1/2+a+it)|\le Ce^{-a\ell}\{L+|P_{Z_\chi}(1/2+a_0+it)|\}. \tag*{(160)} ∣ L χ ( 1/2 + a + i t ) − P χ ( 1/2 + a + i t ) ∣ ≤ C e − a ℓ { L + ∣ P Z χ ( 1/2 + a 0 + i t ) ∣ } . (160) This step uses neither a ± ℑ ρ \pm\Im\rho ± ℑ ρ L χ \mathcal{L}_\chi L χ
For complex a a a (160) to its real and imaginary parts and then use (156) . The precise normalized bound is
∥ a ( L χ − P χ ) ( 1 / 2 + a ) ∥ q ≤ C q e − ℜ ( a ) ℓ ( ∣ a ∣ L + ∣ a ∣ / a 0 ) . \lVert a(\mathcal{L}_\chi-P_\chi)(1/2+a)\rVert_q\le C_qe^{-\Re(a)\ell}(|a|L+|a|/a_0). ∥ a ( L χ − P χ ) ( 1/2 + a ) ∥ q ≤ C q e − ℜ ( a ) ℓ ( ∣ a ∣ L + ∣ a ∣/ a 0 ) . Put x = ℜ ( a ) ℓ x=\Re(a)\ell x = ℜ ( a ) ℓ ∣ a ∣ ≪ ℜ a |a|\ll\Re a ∣ a ∣ ≪ ℜ a x ≥ C y H x\ge C_yH x ≥ C y H L / ℓ = H 120 / C y L/\ell=H^{120}/C_y L / ℓ = H 120 / C y C q H 120 x e − x C_qH^{120}xe^{-x} C q H 120 x e − x (157) . Keeping the product x e − x xe^{-x} x e − x (156) are a P χ aP_{\chi} a P χ a 0 P Z χ a_{0}P_{Z_{\chi}} a 0 P Z χ 1 / ∣ a ∣ 1/|a| 1/∣ a ∣ ℓ \ell ℓ ℜ ( a ) ℓ → ∞ \Re(a)\ell\to\infty ℜ ( a ) ℓ → ∞
Lemma 7.5 (Small balls for the arithmetic complex process). Uniformly for t t t (152) ,
P X { ∣ G χ ( t ) ∣ ≤ H − 5 } ≪ H − 10 . (161) \mathbb{P}_{X}\{|G_{\chi}(t)|\le H^{-5}\}\ll H^{-10}. \tag*{(161)} P X { ∣ G χ ( t ) ∣ ≤ H − 5 } ≪ H − 10 . (161) Proof. Set r = H − 5 r=H^{-5} r = H − 5 f r ( x , y ) = e 2 exp ( − ( x 2 + y 2 ) / r 2 ) f_r(x,y)=e^2\exp(-(x^2+y^2)/r^2) f r ( x , y ) = e 2 exp ( − ( x 2 + y 2 ) / r 2 ) ∣ x ∣ , ∣ y ∣ ≤ r |x|,|y|\le r ∣ x ∣ , ∣ y ∣ ≤ r f ^ ( τ ) = ∫ e − i τ ⋅ x f ( x ) d x \widehat{f}(\tau)=\int e^{-i\tau\cdot x}f(x)\,dx f ( τ ) = ∫ e − i τ ⋅ x f ( x ) d x
( 2 π ) − 2 ∥ f r ^ ∥ 1 = e 2 , ( 2 π ) − 2 ∫ ∥ τ ∥ ∞ > T F ∣ f r ^ ( τ ) ∣ d τ ≪ e − c ( r T F ) 2 . (2\pi)^{-2}\|\widehat{f_r}\|_1=e^2,\qquad(2\pi)^{-2}\int_{\|\tau\|_{\infty}>T_F}|\widehat{f_r}(\tau)|\,d\tau\ll e^{-c(rT_F)^2}. ( 2 π ) − 2 ∥ f r ∥ 1 = e 2 , ( 2 π ) − 2 ∫ ∥ τ ∥ ∞ > T F ∣ f r ( τ ) ∣ d τ ≪ e − c ( r T F ) 2 . Use T F = H 50 T_F=H^{50} T F = H 50 M = ⌈ H 102 ⌉ M=\lceil H^{102}\rceil M = ⌈ H 102 ⌉ Lemma 7.4 . Then
T F / M = O ( H − 1 ) , M ℓ / L = O ( H − 18 ) , M log ( 2 + T F ) = o ( L ) . T_F/\sqrt{M}=O(H^{-1}),\qquad M\ell/L=O(H^{-18}),\qquad M\log(2+T_F)=o(L). T F / M = O ( H − 1 ) , M ℓ / L = O ( H − 18 ) , M log ( 2 + T F ) = o ( L ) . Fourier inversion and Lemma 7.3 compare the arithmetic and random polynomials with error smaller than every fixed inverse power of H H H Lemma 7.4 , multiplied by the Lipschitz norm O ( r − 1 ) O(r^{-1}) O ( r − 1 ) C ( 1 + ∣ a ∣ ℓ ) e − c ℜ ( a ) ℓ + C e − c ℓ C(1+|a|\ell)e^{-c\Re(a)\ell}+Ce^{-c\ell} C ( 1 + ∣ a ∣ ℓ ) e − c ℜ ( a ) ℓ + C e − c ℓ
Finally a ≤ H − 50 a\le H^{-50} a ≤ H − 50 f r f_r f r O ( r − j ) O(r^{-j}) O ( r − j ) O ( H − 45 ) O(H^{-45}) O ( H − 45 ) N 2 ( 0 , 1 8 I 2 ) N_2(0,\frac18 I_2) N 2 ( 0 , 8 1 I 2 ) E f r ≪ r 2 \mathbb{E}f_r\ll r^2 E f r ≪ r 2 f r f_r f r (161) . □
A grid argument and domination by the real product
Lemma 7.6 (Bulk exclusion). The probability that G χ G_{\chi} G χ I H = [ 50 log H , H − 121 log H ] I_H=[50\log H,H-121\log H] I H = [ 50 log H , H − 121 log H ] O ( H − 3 ) O(H^{-3}) O ( H − 3 )
Proof. Cover I H I_H I H O ( H ) O(H) O ( H ) Lemma 7.4 . Markov’s inequality and a union bound give
P X { sup t ∈ I H ∣ G χ ′ ( t ) ∣ > H } ≪ H − 15 . \mathbb{P}_{X}\left\{\sup_{t\in I_H}|G_{\chi}'(t)|>H\right\}\ll H^{-15}. P X { t ∈ I H sup ∣ G χ ′ ( t ) ∣ > H } ≪ H − 15 . A deterministic grid of spacing at most h = H − 6 h=H^{-6} h = H − 6 O ( H 7 ) O(H^7) O ( H 7 ) G χ ( t 0 ) = 0 G_{\chi}(t_0)=0 G χ ( t 0 ) = 0 H H H h h h ∣ G χ ( t j ) ∣ ≤ H h = H − 5 |G_{\chi}(t_j)|\le Hh=H^{-5} ∣ G χ ( t j ) ∣ ≤ H h = H − 5 Lemma 7.5 and another union bound give O ( H − 15 ) + O ( H 7 H − 10 ) = O ( H − 3 ) O(H^{-15})+O(H^7H^{-10})=O(H^{-3}) O ( H − 15 ) + O ( H 7 H − 10 ) = O ( H − 3 ) L ( 1 / 2 + e − t , χ ) L(1/2+e^{-t},\chi) L ( 1/2 + e − t , χ ) t t t L ′ ( 1 / 2 + e − t , χ ) L'(1/2+e^{-t},\chi) L ′ ( 1/2 + e − t , χ )
Lemma 7.7 (Deterministic domination on the real axis). Let f f f J J J f # ( s ) = f ( s ‾ ) ‾ f^{\#}(s)=\overline{f(\overline{s})} f # ( s ) = f ( s ) ( f f # ) ′ (ff^{\#})' ( f f # ) ′
R f ′ m u l t ( J ) ≤ R ( f f # ) ′ m u l t ( J ) . (162) R_{f'}^{\mathrm{mult}}(J)\le R_{(ff^{\#})'}^{\mathrm{mult}}(J). \tag*{(162)} R f ′ mult ( J ) ≤ R ( f f # ) ′ mult ( J ) . (162) Proof. At a real point x x x f ( x ) ≠ 0 f(x)\ne0 f ( x ) = 0 f ′ f' f ′ k k k ( f f # ) ′ (ff^{\#})' ( f f # ) ′ k k k ( f f # ) ′ = f ′ f # + f ( f # ) ′ (ff^{\#})'=f'f^{\#}+f(f^{\#})' ( f f # ) ′ = f ′ f # + f ( f # ) ′ k k k f f f m ≥ 1 m\ge1 m ≥ 1 x x x f # f^{\#} f # f f # ff^{\#} f f # 2 m 2m 2 m 2 m − 1 ≥ m − 1 2m-1\ge m-1 2 m − 1 ≥ m − 1 f ′ f' f ′
Proof of Theorem 7.1. Take f = L ( ⋅ , χ ) f=L(\cdot,\chi) f = L ( ⋅ , χ ) f # = L ( ⋅ , χ ˉ ) f^\#=L(\cdot,\bar{\chi}) f # = L ( ⋅ , χ ˉ ) Z χ = f f # Z_\chi=ff^\# Z χ = f f # q χ q_\chi q χ − 1 -1 − 1 Z χ Z_\chi Z χ
The front interval in the t t t [ log 2 , 50 log H ] [\log2,50\log H] [ log 2 , 50 log H ] O ( log H ) O(\log H) O ( log H ) Z χ Z_\chi Z χ O ( log H ) O(\log H) O ( log H ) R X R_X R X
E X [ 1 R X R Z χ m u l t ( f r o n t ) ] ≪ log H , P X ( R X c ) ≪ H − 7 . \mathbb{E}_X\left[1_{R_X}R_{Z_\chi}^{\mathrm{mult}}(\mathrm{front})\right]\ll\log H,\qquad\mathbb{P}_X(R_X^c)\ll H^{-7}. E X [ 1 R X R Z χ mult ( front ) ] ≪ log H , P X ( R X c ) ≪ H − 7 . By Markov and domination, the front count for f ′ f' f ′ H 1 / 2 + ε H^{1/2+\varepsilon} H 1/2 + ε C ( log H ) H − 1 / 2 − ε + C H − 7 C(\log H)H^{-1/2-\varepsilon}+CH^{-7} C ( log H ) H − 1/2 − ε + C H − 7 t → ∞ t\to\infty t → ∞
The left-hand interval 0 ≤ s − 1 / 2 ≤ H 1 / 2 / L 0\le s-1/2\le H^{1/2}/L 0 ≤ s − 1/2 ≤ H 1/2 / L
ν = H 1 / 2 + ε / 4 , ℓ c = L / H 1 / 2 + ε / 8 , q = ⌈ 8 / ε ⌉ , z = H − 1 / 2 + ε . \nu=H^{1/2+\varepsilon/4},\qquad\ell_c=L/H^{1/2+\varepsilon/8},\qquad q=\lceil8/\varepsilon\rceil,\qquad z=H^{-1/2+\varepsilon}. ν = H 1/2 + ε /4 , ℓ c = L / H 1/2 + ε /8 , q = ⌈ 8/ ε ⌉ , z = H − 1/2 + ε . Then z H = H 1 / 2 + ε zH=H^{1/2+\varepsilon} z H = H 1/2 + ε ( ν / ( z H ) ) q ≤ H − 6 (\nu/(zH))^q\le H^{-6} ( ν / ( z H ) ) q ≤ H − 6 C ε H 1 / 2 + ε C_\varepsilon H^{1/2+\varepsilon} C ε H 1/2 + ε O ε ( H − 2 log H + H − 6 ) O_\varepsilon(H^{-2}\log H+H^{-6}) O ε ( H − 2 log H + H − 6 ) q χ 2 q_\chi^2 q χ 2 log log ( q χ 2 ) = H χ + log 2 \log\log(q_\chi^2)=H_\chi+\log2 log log ( q χ 2 ) = H χ + log 2 H H H f ′ f' f ′
The intervening bulk has no real zeros except on the event in Lemma 7.6. Adding the three failure probabilities and allowing overlap at the boundaries proves (151) , since H χ = H + O ( 1 / log X ) H_\chi=H+O(1/\log X) H χ = H + O ( 1/ log X )
For q χ ≤ X q_\chi\le X q χ ≤ X q χ ≤ X q_\chi\le\sqrt{X} q χ ≤ X O ( X ) O(\sqrt{X}) O ( X ) (145) . On all remaining dyadic blocks log log q χ = H + O ( 1 ) \log\log q_\chi=H+O(1) log log q χ = H + O ( 1 ) O ( X ) O(X) O ( X ) □ \square □
Remark 7.8. Theorem 7.1 is a probability bound for the original family. It does not assert absence of real zeros on the whole interval, a bounded total count, or a mean bound obtained by integrating an uncontrolled exceptional tail. In particular its conclusion should not be written as an asymptotic with zero leading constant. The positive intensity 1 / ( π 2 ) 1/(\pi\sqrt{2}) 1/ ( π 2 ) Z χ ′ Z'_\chi Z χ ′
Numerical diagnostics and reproducibility
The computations in this section are diagnostics, not inputs to a proof. All plotted coordinates are extracted from saved result files. The source archive contains the CSV files and their SHA–256 provenance manifest. Sampling errors and numerical discretization errors are different quantities; we report both where relevant. A finite arithmetic census has no sampling standard error, but still has numerical evaluation and missed-root risks.
The Gaussian crossing calculation
For the Gaussian process with mean and covariance corresponding to θ = 1 \theta=1 θ = 1 Figure 4 compares the two calculations. The same samples were used at different mesh widths, so the displayed estimates are correlated across widths. At h = 0.005 h=0.005 h = 0.005
Figure 4. A check of the Gaussian two-point crossing calculation. This is Gaussian data, not actual arithmetic or an Euler-product simulation. The finite-mesh bias and Monte Carlo standard error are both visible.
Local quadratic distributions and finite root counts
The quadratic computation evaluated all 60 786 nonprincipal fundamental discriminants with ∣ D ∣ ≤ 10 5 |D| \le10^{5} ∣ D ∣ ≤ 1 0 5 χ D ( 2 ) = 1 \chi_D(2)=1 χ D ( 2 ) = 1 10 6 10^{6} 1 0 6 Figure 5 uses δ = 1 / 2 \delta=1/2 δ = 1/2 F ^ ( 1 − F ^ ) / 20000 ≤ 0.00354 \sqrt{\widehat{F}(1-\widehat{F})/20000} \le0.00354 F ( 1 − F ) /20000 ≤ 0.00354
Figure 5. Local distributions at δ = 1 / 2 \delta=1/2 δ = 1/2 χ D ( 2 ) = X ( 2 ) = 1 \chi_D(2)=X(2)=1 χ D ( 2 ) = X ( 2 ) = 1
The empirical root counts in Figure 6 agreed on meshes with 64, 128, and 512 subdivisions and in two winding checks. Selected values were also recomputed at higher precision. These agreements do not certify all roots or their multiplicities. There are 30 394 positive and 30 392 negative discriminants. Their recorded mean counts are 0.27406725 and 0.31248355, respectively. The upper bound log log ( 10 5 ) \log\log(10^5) log log ( 1 0 5 )
Figure 6. Empirical counts for ∣ D ∣ ≤ 10 5 |D| \le10^5 ∣ D ∣ ≤ 1 0 5 log log \log\log log log c B M c_{\mathrm{BM}} c BM
An actual modular twist
For the normalized weight-12 form Δ \Delta Δ Figure 7 shows D = 17 D = 17 D = 17 17 2 = 289 17^2 = 289 1 7 2 = 289 2.42 ⋅ 10 − 27 2.42 \cdot10^{-27} 2.42 ⋅ 1 0 − 27 L L L 6.53 ⋅ 10 − 27 6.53 \cdot10^{-27} 6.53 ⋅ 1 0 − 27 L ′ L' L ′
Figure 7. An actual modular twist, sampled at 129 real arguments. The derivative changes sign between 1 / 2 1/2 1/2 1 / 2 + 1 / 128 1/2 + 1/128 1/2 + 1/128
The cost of eight fixed anchors
Positive definiteness of the two fixed Gram matrices used in the proof is analytic, but their conditioning explains the size of possible constants. For the central nodes v j = − log ( 1 + j / 9 ) v_j=-\log(1+j/9) v j = − log ( 1 + j /9 ) v j = − 1 / 8 + j / 36 v_j=-1/8+j/36 v j = − 1/8 + j /36 1 ≤ j ≤ 8 1\le j\le8 1 ≤ j ≤ 8 r ( v i − v j ) r(v_i-v_j) r ( v i − v j ) 5.2463 ⋅ 10 − 16 5.2463\cdot10^{-16} 5.2463 ⋅ 1 0 − 16 5.8477 ⋅ 10 − 22 5.8477\cdot10^{-22} 5.8477 ⋅ 1 0 − 22 Figure 8 uses the saved 260-digit calculation. Comparison with 180 digits gave relative changes below 10 − 159 10^{-159} 1 0 − 159 L D L T LDL^{\mathsf T} L D L T
Figure 8. Eigenvalues of the two fixed 8 × 8 8\times8 8 × 8
As a separate exact algebra check, enumeration of primitive cubic characters up to conductor 5000 gave 1590 oriented characters and 795 fields. The conductor-9 decomposition and mixed local words were checked with exact arithmetic. Such finite checks detect normalization or conjugation errors; they do not prove the asymptotic orthogonality estimate and do not evaluate actual cubic L L L
Assembly of the abstract order-of-magnitude theorem
We prove the ideal-family theorem stated with the exact axioms in the transfer section. This argument is included separately because it yields an exponentially small exceptional proportion for order-of-magnitude bounds, whereas the exact arithmetic intensities require the more precise central arguments above. We retain the notation and constants of those axioms. In particular, the moment for a mollified function is an input, and is not inferred from a functional equation.
Let K K K
∥ P ∥ a r , λ = ∑ a ∣ c a ∣ ( N a ) λ . \lVert P\rVert_{\mathrm{ar},\lambda}=\sum_{\mathfrak a}|c_{\mathfrak a}|(N\mathfrak a)^\lambda. ∥ P ∥ ar , λ = a ∑ ∣ c a ∣ ( N a ) λ . It is submultiplicative by ideal convolution. The elementary bound # { a : N a = n } ≤ d [ K : Q ] ( n ) \#\{\mathfrak a:N\mathfrak a=n\}\le d_{[K:\mathbb Q]}(n) # { a : N a = n } ≤ d [ K : Q ] ( n ) ∑ N a ≤ Y 1 ≤ Y ( 1 + log Y ) [ K : Q ] − 1 \sum_{N\mathfrak a\le Y}1\le Y(1+\log Y)^{[K:\mathbb Q]-1} ∑ N a ≤ Y 1 ≤ Y ( 1 + log Y ) [ K : Q ] − 1 ℓ \ell ℓ C K e C K , ℓ C_K e^{C_{K,\ell}} C K e C K , ℓ X − κ o X^{-\kappa_o} X − κ o
The zero detector and the off-line anchor
Write L X = log X L_X=\log X L X = log X H = log L X H=\log L_X H = log L X 0 ≤ v ≤ 3 / 2 0\le v\le3/2 0 ≤ v ≤ 3/2 t t t
E ∣ L D ( 1 / 2 + v + i t ) M D ( 1 / 2 + v + i t ) − 1 ∣ 2 ≪ L X C m ( 1 + ∣ t ∣ ) C t e − κ m v L X . \mathbb{E}\left|L_D(1/2+v+it)M_D(1/2+v+it)-1\right|^2\ll L_X^{C_m}(1+|t|)^{C_t}e^{-\kappa_m vL_X}. E ∣ L D ( 1/2 + v + i t ) M D ( 1/2 + v + i t ) − 1 ∣ 2 ≪ L X C m ( 1 + ∣ t ∣ ) C t e − κ m v L X . The mollifier is holomorphic in the region in question. If ρ = 1 / 2 + b + i γ \rho=1/2+b+i\gamma ρ = 1/2 + b + iγ − 1 -1 − 1 ρ \rho ρ a ≤ b ≤ 2 a a\le b\le2a a ≤ b ≤ 2 a T a = T ∗ + ℓ − 1 e 3 ℓ a T_a=T_*+\ell^{-1}e^{3\ell a} T a = T ∗ + ℓ − 1 e 3 ℓ a a / 4 a/4 a /4
16 π a 2 ∫ 3 a / 4 9 a / 4 ∫ − T a − a / 4 T a + a / 4 ∣ L D M D − 1 ∣ 2 d t d v . \frac{16}{\pi a^2}\int_{3a/4}^{9a/4}\int_{-T_a-a/4}^{T_a+a/4}\left|L_D M_D-1\right|^2\,dt\,dv. π a 2 16 ∫ 3 a /4 9 a /4 ∫ − T a − a /4 T a + a /4 ∣ L D M D − 1 ∣ 2 d t d v . The rectangle is deterministic and lies in the stated shift range. Taking its expectation gives
≪ L X C m a − 1 ( 1 + T a ) C t + 1 e − 3 κ m a L X / 4 . \ll L_X^{C_m}a^{-1}(1+T_a)^{C_t+1}e^{-3\kappa_m aL_X/4}. ≪ L X C m a − 1 ( 1 + T a ) C t + 1 e − 3 κ m a L X /4 . For a ≥ L X − 1 a\ge L_X^{-1} a ≥ L X − 1 T ∗ ≤ 1 T_*\le1 T ∗ ≤ 1 8 ≤ ℓ ≤ c ℓ L X 8\le\ell\le c_\ell L_X 8 ≤ ℓ ≤ c ℓ L X c ℓ ≤ κ m / [ 16 ( C t + 1 ) ] c_\ell\le\kappa_m/[16(C_t+1)] c ℓ ≤ κ m / [ 16 ( C t + 1 )]
P { ∃ ρ = 1 / 2 + b + i γ : b > a ∗ , ∣ γ ∣ ≤ T ∗ + ℓ − 1 e 3 ℓ b } ≪ L X C m + 2 e − c z a ∗ L X . (163) \mathbb{P}\{\exists\rho=1/2+b+i\gamma:b>a_*,\ |\gamma|\le T_*+\ell^{-1}e^{3\ell b}\}\ll L_X^{C_m+2}e^{-c_za_*L_X}. \tag*{(163)} P { ∃ ρ = 1/2 + b + iγ : b > a ∗ , ∣ γ ∣ ≤ T ∗ + ℓ − 1 e 3 ℓ b } ≪ L X C m + 2 e − c z a ∗ L X . (163) Here L X − 1 ≤ a ∗ ≤ 1 / 2 L_X^{-1}\le a_*\le1/2 L X − 1 ≤ a ∗ ≤ 1/2
We record the absorption calculation in the presence of off-line zeros. Suppose 4 / ℓ ≤ a ≤ 1 4/\ell\le a\le1 4/ ℓ ≤ a ≤ 1 ∣ t ∣ ≤ 1 |t|\le1 ∣ t ∣ ≤ 1 b > a / 2 b>a/2 b > a /2 ∣ γ − t ∣ ≤ ℓ − 1 e 3 ℓ b |\gamma-t|\le\ell^{-1}e^{3\ell b} ∣ γ − t ∣ ≤ ℓ − 1 e 3 ℓ b 1 / 2 ± b + i γ 1/2\pm b+i\gamma 1/2 ± b + iγ ∑ a / ∣ 1 / 2 + a + i t − ρ ∣ 2 \sum a/|1/2+a+it-\rho|^2 ∑ a /∣1/2 + a + i t − ρ ∣ 2
a 2 + ( t − γ ) 2 − b 2 a 2 + ( t − γ ) 2 + b 2 ≥ 3 5 . \frac{a^2+(t-\gamma)^2-b^2}{a^2+(t-\gamma)^2+b^2}\ge\frac35. a 2 + ( t − γ ) 2 + b 2 a 2 + ( t − γ ) 2 − b 2 ≥ 5 3 . For b ≤ a / 2 b\le a/2 b ≤ a /2 ℓ b > 2 \ell b>2 ℓ b > 2 ∣ γ − t ∣ > 2 b |\gamma-t|>2b ∣ γ − t ∣ > 2 b A = ℓ a A=\ell a A = ℓ a B = ℓ b B=\ell b B = ℓ b C = ℓ ( t − γ ) C=\ell(t-\gamma) C = ℓ ( t − γ )
e B − A + 2 e 2 ( B − A ) + e 3 ( B − A ) A ( A − B ) 2 + C 2 . \frac{e^{B-A}+2e^{2(B-A)}+e^{3(B-A)}}{A\sqrt{(A-B)^2+C^2}}. A ( A − B ) 2 + C 2 e B − A + 2 e 2 ( B − A ) + e 3 ( B − A ) . If B ≤ A / 2 B\le A/2 B ≤ A /2 2 A − 2 ( e − A / 2 + 2 e − A + e − 3 A / 2 ) < 1 / 32 2A^{-2}(e^{-A/2}+2e^{-A}+e^{-3A/2})<1/32 2 A − 2 ( e − A /2 + 2 e − A + e − 3 A /2 ) < 1/32 A ≥ 4 A\ge4 A ≥ 4 B > A / 2 B>A/2 B > A /2 ∣ C ∣ > e 3 B |C|>e^{3B} ∣ C ∣ > e 3 B A − 1 ( e − 2 A + 2 e − 5 A / 2 + e − 3 A ) < 1 / 32 A^{-1}(e^{-2A}+2e^{-5A/2}+e^{-3A})<1/32 A − 1 ( e − 2 A + 2 e − 5 A /2 + e − 3 A ) < 1/32 s = 1 / 2 + a 1 + i t s=1/2+a_1+it s = 1/2 + a 1 + i t a ≤ a 1 ≤ 3 / 2 a\le a_1\le3/2 a ≤ a 1 ≤ 3/2 A i = ℓ a i A_i=\ell a_i A i = ℓ a i a 0 = a a_0=a a 0 = a D i 2 = ( A i − B ) 2 + C 2 D_i^2=(A_i-B)^2+C^2 D i 2 = ( A i − B ) 2 + C 2 a 1 a_1 a 1 a 0 a_0 a 0
D 0 2 A 0 D 1 3 { e B − A 1 + 2 e 2 ( B − A 1 ) + e 3 ( B − A 1 ) } . \frac{D_0^2}{A_0D_1^3}\left\{e^{B-A_1}+2e^{2(B-A_1)}+e^{3(B-A_1)}\right\}. A 0 D 1 3 D 0 2 { e B − A 1 + 2 e 2 ( B − A 1 ) + e 3 ( B − A 1 ) } . If B ≤ A 0 / 2 B\le A_0/2 B ≤ A 0 /2 D 1 ≥ D 0 D_1\ge D_0 D 1 ≥ D 0 D 1 ≥ A 1 / 2 D_1\ge A_1/2 D 1 ≥ A 1 /2 B > A 0 / 2 B>A_0/2 B > A 0 /2 D 0 2 ≤ 2 C 2 D_0^2\le2C^2 D 0 2 ≤ 2 C 2 D 1 ≥ ∣ C ∣ > e 3 B D_1\ge|C|>e^{3B} D 1 ≥ ∣ C ∣ > e 3 B
∣ L D ( s ) − P ℓ , D ( s ) ∣ ≪ K e − ℓ a 1 / 2 { log Q D + ∣ P ℓ , D ( 1 / 2 + a + i t ) ∣ } . (164) \left|L_D(s)-P_{\ell,D}(s)\right|\ll_K e^{-\ell a_1/2}\left\{\log Q_D+\left|P_{\ell,D}(1/2+a+it)\right|\right\}. \tag*{(164)} ∣ L D ( s ) − P ℓ , D ( s ) ∣ ≪ K e − ℓ a 1 /2 { log Q D + ∣ P ℓ , D ( 1/2 + a + i t ) ∣ } . (164) When a = 4 / ℓ a=4/\ell a = 4/ ℓ 1 / 2 1/2 1/2 1 1 1 B ≤ 2 B\le2 B ≤ 2
A lower bound without a discrepancy axiom
In H / 20 ≤ t j ≤ H / 10 H/20 \le t_j \le H/10 H /20 ≤ t j ≤ H /10 d ≪ H d \ll H d ≪ H η 0 = e − H / 200 \eta_0=e^{-H/200} η 0 = e − H /200 ℓ = C L H e H / 10 \ell=C_LHe^{H/10} ℓ = C L H e H /10 T = H / η 0 T=H/\eta_0 T = H / η 0 M = ⌈ H 6 e H / 100 ⌉ M=\lceil H^6e^{H/100}\rceil M = ⌈ H 6 e H /100 ⌉ σ y = 1 / 2 + 4 / ℓ \sigma_y=1/2+4/\ell σ y = 1/2 + 4/ ℓ E ∣ P y ( σ y ) ∣ 2 ≪ ℓ 2 \mathbb{E}|P_y(\sigma_y)|^2\ll\ell^2 E ∣ P y ( σ y ) ∣ 2 ≪ ℓ 2 ∣ P y ( σ y ) ∣ ≤ ℓ e K a H |P_y(\sigma_y)|\le\ell e^{K_aH} ∣ P y ( σ y ) ∣ ≤ ℓ e K a H O ( e − 2 K a H ) O(e^{-2K_aH}) O ( e − 2 K a H ) E E E K a > E / 2 + 1 K_a>E/2+1 K a > E /2 + 1 C L C_L C L O ( e − E H ) O(e^{-EH}) O ( e − E H )
max j ∣ G D ( t j ) − δ j P y ( 1 / 2 + δ j ) ∣ ≤ C max j δ j e − δ j ℓ ( ℓ e K a H + B L X ) ≤ e − E ′ H . \max_j |G_D(t_j)-\delta_jP_y(1/2+\delta_j)|\le C\max_j\delta_je^{-\delta_j\ell}(\ell e^{K_aH}+BL_X)\le e^{-E'H}. j max ∣ G D ( t j ) − δ j P y ( 1/2 + δ j ) ∣ ≤ C j max δ j e − δ j ℓ ( ℓ e K a H + B L X ) ≤ e − E ′ H . where E ′ E' E ′ C L C_L C L L 2 L^2 L 2
X − κ 0 exp { C K , λ M ( ℓ + log H + H / 200 + 1 ) } = O ( X − κ 0 / 2 ) , X^{-\kappa_0}\exp\{C_{K,\lambda}M(\ell+\log H+H/200+1)\}=O(X^{-\kappa_0/2}), X − κ 0 exp { C K , λ M ( ℓ + log H + H /200 + 1 )} = O ( X − κ 0 /2 ) , because M ℓ = O ( H 7 e 11 H / 100 ) = o ( L X ) M\ell=O(H^7e^{11H/100})=o(L_X) M ℓ = O ( H 7 e 11 H /100 ) = o ( L X ) [ − H , H ] [-H,H] [ − H , H ] k = ⌊ c 0 H 2 ⌋ k=\lfloor c_0H^2\rfloor k = ⌊ c 0 H 2 ⌋ c 0 c_0 c 0 k ℓ = o ( L X ) k\ell=o(L_X) k ℓ = o ( L X ) O ( d e − c H 2 ) + O ( e − E H ) O(de^{-cH^2})+O(e^{-EH}) O ( d e − c H 2 ) + O ( e − E H ) I I I η 0 2 \eta_0^2 η 0 2 h I = 1 I ∗ φ η 0 h_I=1_I*\varphi_{\eta_0} h I = 1 I ∗ φ η 0
∣ h I ^ ( τ ) ∣ ≤ min ( ∣ I ∣ , 2 / ∣ τ ∣ ) e − η 0 2 τ 2 / 2 , ( 2 π ) − 1 ∥ h I ^ ∥ 1 ≤ C H . |\widehat{h_I}(\tau)|\le\min(|I|,2/|\tau|)e^{-\eta_0^2\tau^2/2},\qquad(2\pi)^{-1}\|\widehat{h_I}\|_1\le CH. ∣ h I ( τ ) ∣ ≤ min ( ∣ I ∣ , 2/∣ τ ∣ ) e − η 0 2 τ 2 /2 , ( 2 π ) − 1 ∥ h I ∥ 1 ≤ C H . The product test has normalized Fourier norm at most ( C H ) d (CH)^d ( C H ) d e − H 2 / 8 e^{-H^2/8} e − H 2 /8 ∣ τ j ∣ ≤ H / η 0 |\tau_j|\le H/\eta_0 ∣ τ j ∣ ≤ H / η 0 M j ≤ C j η 0 − j M_j\le C_j\eta_0^{-j} M j ≤ C j η 0 − j b ≤ e − H / 20 b\le e^{-H/20} b ≤ e − H /20 d j d^j d j η 0 − j \eta_0^{-j} η 0 − j r = H η 0 r=H\eta_0 r = H η 0 O ( d e − H 2 / 2 ) O(de^{-H^2/2}) O ( d e − H 2 /2 ) O ( d r ) O(dr) O ( d r ) O K ( e − H / 400 ) O_K(e^{-H/400}) O K ( e − H /400 )
Take J = ⌊ κ H ⌋ J=\lfloor\kappa H\rfloor J = ⌊ κ H ⌋ log 4 \log4 log 4 O ( J ) O(J) O ( J ) c J cJ c J θ \theta θ c ′ J c'J c ′ J e − c ′ ′ J e^{-c''J} e − c ′′ J 9 J 9^J 9 J κ \kappa κ κ log 9 < 1 / 800 \kappa\log9<1/800 κ log 9 < 1/800 L D L_D L D L D ′ L'_D L D ′ R D d i s t ≥ c H R_D^{\mathrm{dist}}\ge cH R D dist ≥ cH O ( e − c H ) O(e^{-cH}) O ( e − cH )
Bulk growth with one fixed choice of constants
Fix δ = A H / L X \delta=AH/L_X δ = A H / L X T X = log ( 1 / δ ) T_X=\log(1/\delta) T X = log ( 1/ δ ) t j = log 2 + j / 2 t_j=\log2+j/2 t j = log 2 + j /2 0 ≤ j ≤ ⌈ 2 ( T X − log 2 ) ⌉ 0\le j\le\lceil2(T_X-\log2)\rceil 0 ≤ j ≤ ⌈ 2 ( T X − log 2 )⌉ t j t_j t j W j , U j , W j ′ W_j,U_j,W_j' W j , U j , W j ′ 1 / 8 , 3 / 4 , 4 / 5 1/8,3/4,4/5 1/8 , 3/4 , 4/5 δ j = e − t j \delta_j=e^{-t_j} δ j = e − t j c R = e 4 / 5 sin ( 4 / 5 ) c_R=e^{4/5}\sin(4/5) c R = e 4/5 sin ( 4/5 ) d R = e 4 / 5 sin ( 4 / 5 ) d_R=e^{4/5}\sin(4/5) d R = e 4/5 sin ( 4/5 ) W j W_j W j c R δ j c_R\delta_j c R δ j d R δ j < 1 d_R\delta_j<1 d R δ j < 1
Choose a sufficiently large fixed B + B_+ B +
b j = B + ( 1 + log + ( δ j / δ ) ) , ℓ j = b j / δ j . b_j=B_+(1+\log^+(\delta_j/\delta)),\qquad\ell_j=b_j/\delta_j. b j = B + ( 1 + log + ( δ j / δ )) , ℓ j = b j / δ j . Then ℓ max ≤ e 1 / 2 B + / δ \ell_{\max}\le e^{1/2}B_+/\delta ℓ m a x ≤ e 1/2 B + / δ ∑ j ( 1 + b j ) = O ( H 2 ) \sum_j(1+b_j)=O(H^2) ∑ j ( 1 + b j ) = O ( H 2 ) ( 2 / ℓ j , ℓ j , d R δ j ) (2/\ell_j,\ell_j,d_R\delta_j) ( 2/ ℓ j , ℓ j , d R δ j ) O ( H exp ( ( C m + 2 − c A / B + ) H ) ) O(H\exp((C_m+2-cA/B_+)H)) O ( H exp (( C m + 2 − c A / B + ) H )) e − 2 H e^{-2H} e − 2 H A A A B + B_+ B +
∑ p ∣ a j , p a k , p ∣ ≪ K δ j δ k ( δ j + δ k ) 2 ≪ K e − ∣ t j − t k ∣ . \sum_{\mathfrak p}|a_{j,\mathfrak p}a_{k,\mathfrak p}|\ll_K \frac{\delta_j\delta_k}{(\delta_j+\delta_k)^2}\ll_K e^{-|t_j-t_k|}. p ∑ ∣ a j , p a k , p ∣ ≪ K ( δ j + δ k ) 2 δ j δ k ≪ K e − ∣ t j − t k ∣ . Its row sum is bounded. For any real or imaginary sign choice, pairing of independent symmetric prime variables bounds the 2 m 2m 2 m d ≤ 3 H d\le3H d ≤ 3 H ( C m d ) m (Cmd)^m ( C m d ) m X − κ 0 exp ( C m ( log ( d + 2 ) + C K , λ ℓ max ) ) X^{-\kappa_0}\exp(Cm(\log(d+2)+C_{K,\lambda}\ell_{\max})) X − κ 0 exp ( C m ( log ( d + 2 ) + C K , λ ℓ m a x )) W j W_j W j U j U_j U j 16 16 16 4 d 4^d 4 d
E ( ∑ j max U j ∣ P j , D p r i m e ∣ ) 2 m ≤ 16 2 m 4 d { ( C m d ) m + X − κ 0 e C m ( log ( d + 2 ) + C K , λ ℓ max ) } . \mathbb{E}\left(\sum_j\max_{U_j}|P^{\mathrm{prime}}_{j,D}|\right)^{2m}\le16^{2m}4^d\left\{(Cmd)^m+X^{-\kappa_0}e^{Cm(\log(d+2)+C_{K,\lambda}\ell_{\max})}\right\}. E ( j ∑ U j max ∣ P j , D prime ∣ ) 2 m ≤ 1 6 2 m 4 d { ( C m d ) m + X − κ 0 e C m ( l o g ( d + 2 ) + C K , λ ℓ m a x ) } . Choose m = ⌊ κ + + H ⌋ m=\lfloor\kappa_++H\rfloor m = ⌊ κ + + H ⌋ κ + \kappa_+ κ + m ℓ max ≪ κ + ( B + / A ) L X m\ell_{\max}\ll\kappa_+(B_+/A)L_X m ℓ m a x ≪ κ + ( B + / A ) L X K P H K_PH K P H K P K_P K P O ( H ) O(H) O ( H )
Cover each auxiliary anchor segment by O ( 1 + b j ) O(1+b_j) O ( 1 + b j ) 1 / ( 2 ℓ j ) 1/(2\ell_j) 1/ ( 2 ℓ j ) 3 / ℓ j 3/\ell_j 3/ ℓ j P / ℓ j P/\ell_j P / ℓ j O ( H 2 ) O(H^2) O ( H 2 ) sup ∣ P ℓ j , D ∣ ≤ H ℓ j \sup|P_{\ell_j,D}|\le H\ell_j sup ∣ P ℓ j , D ∣ ≤ H ℓ j
max U j ∣ G D − P j , D ∣ ≪ H e − c R B + ( 1 + log + r j ) { B + ( 1 + log + r j ) + C A r j } , r j = δ j / δ . \max_{U_j}|G_D-P_{j,D}|\ll He^{-c_RB_+(1+\log^+r_j)}\{B_+(1+\log^+r_j)+CAr_j\},\qquad r_j=\delta_j/\delta. U j max ∣ G D − P j , D ∣ ≪ H e − c R B + ( 1 + l o g + r j ) { B + ( 1 + log + r j ) + C A r j } , r j = δ j / δ . Because the r j r_j r j c R B + > 2 c_RB_+>2 c R B + > 2 O A ( H ) O_A(H) O A ( H ) A X \mathcal A_X A X O ( e − c H ) O(e^{-cH}) O ( e − cH )
∑ j M j , D ≤ K + ( A ) H , M j , D = max U j ∣ G D ∣ . (165) \sum_j M_{j,D}\le K_+(A)H,\qquad M_{j,D}=\max_{U_j}|G_D|. \tag*{(165)} j ∑ M j , D ≤ K + ( A ) H , M j , D = U j max ∣ G D ∣. (165) Retained small values and the tip coefficient vector
Put u = ⌊ H ⌋ u=\lfloor\sqrt H\rfloor u = ⌊ H ⌋ δ c u t = H 8 e 2 u / L X \delta_{\mathrm{cut}}=H^8e^{2u}/L_X δ cut = H 8 e 2 u / L X δ j ≥ δ c u t \delta_j\ge\delta_{\mathrm{cut}} δ j ≥ δ cut 1 ≤ n j ≤ u 1\le n_j\le u 1 ≤ n j ≤ u
∏ j exp ( 1 − e 2 n j x j 2 / 4 ) . \prod_j\exp(1-e^{2n_jx_j^2/4}). j ∏ exp ( 1 − e 2 n j x j 2 /4 ) . Writing a factor as g n ( x ) g_n(x) g n ( x ) g n ( x ) ≤ C a 0 e − a 0 n ∣ x ∣ − a 0 g_n(x)\le C_{a_0}e^{-a_0n|x|^{-a_0}} g n ( x ) ≤ C a 0 e − a 0 n ∣ x ∣ − a 0 ∣ x j ∣ ≤ 2 e − n j |x_j|\le2e^{-n_j} ∣ x j ∣ ≤ 2 e − n j C e d ∑ j e n j C e^d\sum_j e^{n_j} C e d ∑ j e n j C s d e − a 0 ∑ j n j C_s^d e^{-a_0\sum_j n_j} C s d e − a 0 ∑ j n j e d e^d e d ∣ τ j ∣ ≤ C E H e n j |\tau_j|\le C_E\sqrt H e^{n_j} ∣ τ j ∣ ≤ C E H e n j C d e d e − c C E 2 H Cde^d e^{-cC_E^2H} C d e d e − c C E 2 H ℓ = C y L X / ( H 7 e 2 u ) \ell=C_yL_X/(H^7e^{2u}) ℓ = C y L X / ( H 7 e 2 u ) M = ⌈ H 4 e 2 u ⌉ M=\lceil H^4e^{2u}\rceil M = ⌈ H 4 e 2 u ⌉ O ( H 3 / 2 e u ) = o ( M ) O(H^{3/2}e^u)=o(\sqrt M) O ( H 3/2 e u ) = o ( M ) M ℓ = O ( L X / H 3 ) M\ell=O(L_X/H^3) M ℓ = O ( L X / H 3 ) M ( u + log H ) = o ( L X ) M(u+\log H)=o(L_X) M ( u + log H ) = o ( L X ) C y C_y C y E > 0 E>0 E > 0
P { A X , ∣ G D ( t j ) ∣ ≤ e − n j ( j ∈ I ) } ≤ C s ∣ I ∣ e − a 0 ∑ j ∈ I n j + O E ( e − E H ) . (166) \mathbb{P}\{\mathcal A_X,\ |G_D(t_j)|\le e^{-n_j}\ (j\in I)\}\le C_s^{|I|}e^{-a_0\sum_{j\in I}n_j}+O_E(e^{-EH}). \tag*{(166)} P { A X , ∣ G D ( t j ) ∣ ≤ e − n j ( j ∈ I )} ≤ C s ∣ I ∣ e − a 0 ∑ j ∈ I n j + O E ( e − E H ) . (166) At an unretained center t t t Δ = e − t \Delta=e^{-t} Δ = e − t S = Δ L X S=\Delta L_X S = Δ L X r = ⌊ ε H ⌋ r=\lfloor\varepsilon\sqrt H\rfloor r = ⌊ ε H ⌋ ε \varepsilon ε S > e − 1 / 2 A H S>e^{-1/2}AH S > e − 1/2 A H Δ ≤ e − H / 2 \Delta\le e^{-H/2} Δ ≤ e − H /2 X X X
a ( z ) = Δ 1 + z 1 − z , H D ( z ) = Δ ( 1 − z ) 2 L D ( 1 / 2 + a ( z ) ) , Z k = 2 θ ( k + 1 ) [ z k ] H D ( z ) , k < r . a(z)=\Delta\frac{1+z}{1-z},\qquad\mathcal H_D(z)=\frac{\Delta}{(1-z)^2}\mathcal L_D(1/2+a(z)),\qquad Z_k=\frac{2}{\sqrt{\theta(k+1)}}[z^k]\mathcal H_D(z),\quad k<r. a ( z ) = Δ 1 − z 1 + z , H D ( z ) = ( 1 − z ) 2 Δ L D ( 1/2 + a ( z )) , Z k = θ ( k + 1 ) 2 [ z k ] H D ( z ) , k < r . For ∣ z ∣ ≤ 1 / 64 |z| \le1/64 ∣ z ∣ ≤ 1/64 w = t − log ( ( 1 + z ) / ( 1 − z ) ) w=t-\log((1+z)/(1-z)) w = t − log (( 1 + z ) / ( 1 − z )) ∣ w − t ∣ < 1 / 16 |w-t|<1/16 ∣ w − t ∣ < 1/16 max V t ∣ G D ∣ ≤ e − u \max_{V_t}|G_D|\le e^{-u} max V t ∣ G D ∣ ≤ e − u ∥ Z ∥ ≤ C r 64 r e − u ≤ 1 / 4 \lVert Z\rVert\le C\sqrt{r}64^r e^{-u}\le1/4 ∥ Z ∥ ≤ C r 6 4 r e − u ≤ 1/4 ϵ log 64 < 1 / 4 \epsilon\log64<1/4 ϵ log 64 < 1/4
The analytic covariance calculation in the random section gives, on ∣ z ∣ , ∣ w ∣ ≤ 1 / 2 |z|,|w|\le1/2 ∣ z ∣ , ∣ w ∣ ≤ 1/2
E H E ( z ) = θ 2 ( 1 − z 2 ) + O ( Δ ) , Cov ( H E ( z ) , H E ( w ) ) = θ 4 ( 1 − z w ) 2 + O ( Δ 2 ) . \mathbb{E}H_E(z)=\frac{\theta}{2(1-z^2)}+O(\Delta),\qquad\operatorname{Cov}(H_E(z),H_E(w))=\frac{\theta}{4(1-zw)^2}+O(\Delta^2). E H E ( z ) = 2 ( 1 − z 2 ) θ + O ( Δ ) , Cov ( H E ( z ) , H E ( w )) = 4 ( 1 − z w ) 2 θ + O ( Δ 2 ) . This covariance is bilinear. Coefficient extraction shows E Z k = θ 1 2 ∣ k / k + 1 + O ( Δ 2 k ) \mathbb{E}Z_k=\sqrt{\theta}\mathbf{1}_{2\mid k}/\sqrt{k+1}+O(\Delta^{2k}) E Z k = θ 1 2 ∣ k / k + 1 + O ( Δ 2 k ) Cov ( Z k , Z l ) = 1 k = l + O ( Δ 2 k + l ) \operatorname{Cov}(Z_k,Z_l)=\mathbf{1}_{k=l}+O(\Delta^{2k+l}) Cov ( Z k , Z l ) = 1 k = l + O ( Δ 2 k + l ) O ( 1 + log ( r + 1 ) ) O(1+\log(r+1)) O ( 1 + log ( r + 1 ))
Set v = r + log ( 2 + S ) v=r+\log(2+S) v = r + log ( 2 + S ) ℓ = C y v / Δ \ell=C_yv/\Delta ℓ = C y v /Δ Y D , k Y_{D,k} Y D , k L D \mathcal{L}_D L D P ℓ , D P_{\ell,D} P ℓ , D ℓ / L X = C y v / S = o ( 1 ) \ell/L_X=C_yv/S=o(1) ℓ / L X = C y v / S = o ( 1 ) ( Δ / 100 , ℓ , 3 Δ ) (\Delta/100,\ell,3\Delta) ( Δ/100 , ℓ , 3Δ ) O ( e − 2 H ) O(e^{-2H}) O ( e − 2 H ) ∣ t ′ ∣ ≤ 3 Δ |t'|\le3\Delta ∣ t ′ ∣ ≤ 3Δ
a D ( t ′ ) = 2 max ( { 2 / ℓ } ∪ { β − 1 / 2 : β > 1 / 2 , ∣ γ − t ′ ∣ ≤ ℓ − 1 3 ℓ ( β − 1 / 2 ) } ) . a_D(t')=2\max\left(\{2/\ell\}\cup\{\beta-1/2:\ \beta>1/2,\ |\gamma-t'|\le\ell^{-1}3\ell(\beta-1/2)\}\right). a D ( t ′ ) = 2 max ( { 2/ ℓ } ∪ { β − 1/2 : β > 1/2 , ∣ γ − t ′ ∣ ≤ ℓ − 1 3 ℓ ( β − 1/2 )} ) . It lies between 4 / ℓ 4/\ell 4/ ℓ Δ / 50 \Delta/50 Δ/50 O ( v 2 ) O(v^2) O ( v 2 ) 1 / ( 2 ℓ ) 1/(2\ell) 1/ ( 2 ℓ ) E ( Δ sup ∣ P ∣ ) 2 ≪ v 4 + o ( 1 ) \mathbb{E}(\Delta\sup|P|)^2\ll v^4+o(1) E ( Δ sup ∣ P ∣ ) 2 ≪ v 4 + o ( 1 ) Δ sup ∣ P ∣ ≤ e b v \Delta\sup|P|\le e^{bv} Δ sup ∣ P ∣ ≤ e b v O ( e − b v ) O(e^{-bv}) O ( e − b v ) b b b ∣ z ∣ ≤ 1 / 2 |z|\le1/2 ∣ z ∣ ≤ 1/2 ℜ a ( z ) ≥ Δ / 3 \Re a(z)\ge\Delta/3 ℜ a ( z ) ≥ Δ/3
sup ∣ H D − H ℓ , D ∣ ≪ e − C y v / 6 ( e b v + C S ) ≤ e − 30 v . \sup|\mathcal{H}_D-\mathcal{H}_{\ell,D}|\ll e^{-C_yv/6}(e^{bv}+CS)\le e^{-30v}. sup ∣ H D − H ℓ , D ∣ ≪ e − C y v /6 ( e b v + C S ) ≤ e − 30 v . after C y C_y C y ∥ Y − Z ∥ ≤ 1 / 4 \lVert Y-Z\rVert\le1/4 ∥ Y − Z ∥ ≤ 1/4 O ( e − c r + e − H ) O(e^{-cr}+e^{-H}) O ( e − cr + e − H ) L 2 L^2 L 2
The finite model vector is Y E = μ + ∑ p [ a p X ( p ) + b p ( X ( p ) 2 − θ p ) ] Y_E=\mu+\sum_p[a_pX(p)+b_p(X(p)^2-\theta_p)] Y E = μ + ∑ p [ a p X ( p ) + b p ( X ( p ) 2 − θ p )] I / 2 I/2 I /2 2 I 2I 2 I X X X e e e ( E ∥ Y E − Z E ∥ 2 ) 1 / 2 (\mathbb{E}\lVert Y_E-Z_E\rVert^2)^{1/2} ( E ∥ Y E − Z E ∥ 2 ) 1/2 Z E Z_E Z E Y D Y_D Y D
∑ p ∥ b p ∥ 2 ≪ K Δ 2 r 4 = o ( 1 ) , β p : = ∥ a p ∥ + ∥ b p ∥ ≪ K Δ r 2 r ( log N p ) ( N p ) − 1 / 2 − Δ / 3 , \sum_p\lVert b_p\rVert^2\ll_K\Delta^2r^4=o(1),\qquad\beta_p:=\lVert a_p\rVert+\lVert b_p\rVert\ll_K\Delta\sqrt{r}\,2^r(\log Np)(Np)^{-1/2-\Delta/3}, p ∑ ∥ b p ∥ 2 ≪ K Δ 2 r 4 = o ( 1 ) , β p := ∥ a p ∥ + ∥ b p ∥ ≪ K Δ r 2 r ( log N p ) ( N p ) − 1/2 − Δ/3 , and so ∑ p β p 3 ≤ Δ 3 e C K r \sum_p\beta_p^3\le\Delta^3e^{C_Kr} ∑ p β p 3 ≤ Δ 3 e C K r max β p ≤ 1 \max\beta_p\le1 max β p ≤ 1 θ p ≥ θ ∗ / 2 \theta_p\ge\theta_*/2 θ p ≥ θ ∗ /2 2 / θ ∗ 2/\theta_* 2/ θ ∗ o ( 1 ) o(1) o ( 1 ) θ p = 0 \theta_p=0 θ p = 0
For m = ⌊ ϵ 0 r ⌋ m=\lfloor\epsilon_0r\rfloor m = ⌊ ϵ 0 r ⌋ A Y = tr Σ + ∥ μ ∥ 2 A_Y=\operatorname{tr}\Sigma+\lVert\mu\rVert^2 A Y = tr Σ + ∥ μ ∥ 2
Q ( y ) = ( A Y − ∥ y ∥ 2 A Y − 1 ) 2 m Q(y)=\left(\frac{A_Y-\lVert y\rVert^2}{A_Y-1}\right)^{2m} Q ( y ) = ( A Y − 1 A Y − ∥ y ∥ 2 ) 2 m is nonnegative and at least one on the unit ball, and its model expectation is O ( e − c r + Δ 3 e C r ) O(e^{-cr}+\Delta^3e^{Cr}) O ( e − cr + Δ 3 e C r ) r r r A Y − 1 ≍ r A_Y-1\asymp r A Y − 1 ≍ r m ≥ 1 m\ge1 m ≥ 1
log ∥ Q ( Y ) ∥ a r , λ ≤ 4 C K , λ m ℓ + O K ( m r + m log ( r + 1 ) ) , m ℓ L X ≪ C y ( ϵ 2 A + ϵ log ( 2 + A H ) A H ) . \log\lVert Q(Y)\rVert_{\mathrm{ar},\lambda}\le4C_{K,\lambda}m\ell+O_K(mr+m\log(r+1)),\qquad\frac{m\ell}{L_X}\ll C_y\left(\frac{\epsilon^2}{A}+\frac{\epsilon\log(2+AH)}{A\sqrt{H}}\right). log ∥ Q ( Y ) ∥ ar , λ ≤ 4 C K , λ m ℓ + O K ( m r + m log ( r + 1 )) , L X m ℓ ≪ C y ( A ϵ 2 + A H ϵ log ( 2 + A H ) ) . Choose A A A 4 m 4m 4 m
sup j i n t i p P { A X , V j max ∣ G D ∣ ≤ e − u } ≪ e − c H . (167) \sup_{j\ \mathrm{in\ tip}}\mathbb{P}\{A_X,V_j\max|G_D|\le e^{-u}\}\ll e^{-c\sqrt{H}}. \tag*{(167)} j in tip sup P { A X , V j max ∣ G D ∣ ≤ e − u } ≪ e − c H . (167) Depth allocation and the bulk upper bound
Let v j = log + ( 1 / m j ) v_j = \log^{+}(1/m_j) v j = log + ( 1/ m j ) m j = max V j ∣ G D ∣ m_j = \max_{V_j} |G_D| m j = max V j ∣ G D ∣ O ( H ) O(H) O ( H ) max v j > u \max v_j > u max v j > u O ( e − c H ) O(e^{-c\sqrt{H}}) O ( e − c H ) q 0 q_0 q 0 q 0 / 2 q_0/2 q 0 /2 q 0 > 16 q_0 > 16 q 0 > 16 3 log C s / q 0 < a 0 / 4 3\log C_s/q_0 < a_0/4 3 log C s / q 0 < a 0 /4 q 0 q_0 q 0 4 u / q 0 + O ( log H ) 4u/q_0 + O(\log H) 4 u / q 0 + O ( log H ) N = ⌈ H ⌉ N = \lceil H\rceil N = ⌈ H ⌉ H / 2 H/2 H /2 H / 2 H/2 H /2 e − γ H e^{-\gamma H} e − γ H γ > 0 \gamma> 0 γ > 0
Refine to classes modulo q = q 1 q 0 q = q_1q_0 q = q 1 q 0 d ∗ = ( 2 / q ) H + O ( log H ) d_* = (2/q)H + O(\log H) d ∗ = ( 2/ q ) H + O ( log H ) N N N d ∗ d_* d ∗
( N + d ∗ − 1 d ∗ − 1 ) ≤ exp ( ( h ( 2 / q ) + o ( 1 ) ) H ) , h ( x ) = ( 1 + x ) log ( 1 + x ) − x log x . \binom{N+d_*-1}{d_*-1} \le\exp((h(2/q)+o(1))H), \qquad h(x)=(1+x)\log(1+x)-x\log x. ( d ∗ − 1 N + d ∗ − 1 ) ≤ exp (( h ( 2/ q ) + o ( 1 )) H ) , h ( x ) = ( 1 + x ) log ( 1 + x ) − x log x . Take q 1 q_1 q 1 h ( 2 / q ) < γ / 2 h(2/q) < \gamma/2 h ( 2/ q ) < γ /2 N + d ∗ N+d_* N + d ∗ N N N ∑ j v j ≤ C v H \sum_j v_j \le C_vH ∑ j v j ≤ C v H O ( e − c H ) O(e^{-c\sqrt{H}}) O ( e − c H )
At a point of maximum modulus in V j V_j V j 1 / 2 1/2 1/2 5 / 8 5/8 5/8 U j U_j U j [ t j − 1 / 4 , t j + 1 / 4 ] [t_j-1/4,t_j+1/4] [ t j − 1/4 , t j + 1/4 ] ( log ( M j / m j ) ) / log ( 5 / 4 ) ≤ ( M j + v j ) / log ( 5 / 4 ) (\log(M_j/m_j))/\log(5/4) \le(M_j+v_j)/\log(5/4) ( log ( M j / m j )) / log ( 5/4 ) ≤ ( M j + v j ) / log ( 5/4 ) m j > 0 m_j > 0 m j > 0 O ( H ) O(H) O ( H ) 1 / 2 + δ 1/2+\delta 1/2 + δ 1 1 1
The central interval and root number
Set s ± = 1 / 2 ± δ s_{\pm}=1/2\pm\delta s ± = 1/2 ± δ ℓ 0 = 4 / δ \ell_0=4/\delta ℓ 0 = 4/ δ ℓ 1 = B 0 / δ \ell_1=B_0/\delta ℓ 1 = B 0 / δ B 0 B_0 B 0
W ℓ , D ( s ) = ∑ p , k ≥ 1 w ℓ ( ( N p ) k ) a D ( p ) k k ( N p ) k s . W_{\ell,D}(s)=\sum_{p,k\geq1}\frac{w_{\ell}((N\mathfrak{p})^k)a_D(\mathfrak{p})^k}{k(N\mathfrak{p})^{ks}}. W ℓ , D ( s ) = p , k ≥ 1 ∑ k ( N p ) k s w ℓ (( N p ) k ) a D ( p ) k . Thus W ℓ , D ′ = − P ℓ , D W'_{\ell,D}=-P_{\ell,D} W ℓ , D ′ = − P ℓ , D W ℓ , D ( s ) → 0 W_{\ell,D}(s)\to0 W ℓ , D ( s ) → 0 s → ∞ s\to\infty s → ∞ L D ( s ) → 1 L_D(s)\to1 L D ( s ) → 1 σ i \sigma_i σ i 2 2 2 2 2 2
log ∣ L D ( σ i ) ∣ = W ℓ i , D ( σ i ) + O ( ( ∣ P ℓ i , D ( σ i ) ∣ + log Q D ) / ℓ i ) , σ i = 1 / 2 + 4 / ℓ i . \log|L_D(\sigma_i)|=W_{\ell_i,D}(\sigma_i)+O\left((|P_{\ell_i,D}(\sigma_i)|+\log Q_D)/\ell_i\right), \qquad\sigma_i=1/2+4/\ell_i. log ∣ L D ( σ i ) ∣ = W ℓ i , D ( σ i ) + O ( ( ∣ P ℓ i , D ( σ i ) ∣ + log Q D ) / ℓ i ) , σ i = 1/2 + 4/ ℓ i . The integrated ideal-series tail beyond 2 2 2 ∑ N a > e ℓ i ( N a ) − 2 ≪ K e − ℓ i ( 1 + ℓ i ) [ K : Q ] − 1 ≪ K e − ℓ i / 2 \sum_{N\mathfrak{a}>e^{\ell_i}}(N\mathfrak{a})^{-2}\ll_K e^{-\ell_i}(1+\ell_i)^{[K:\mathbb{Q}]-1}\ll_K e^{-\ell_i/2} ∑ N a > e ℓ i ( N a ) − 2 ≪ K e − ℓ i ( 1 + ℓ i ) [ K : Q ] − 1 ≪ K e − ℓ i /2
At real s + s_+ s + W W W U U U
E ∣ U ∣ 2 m ≤ ( C m H ) m + X − κ o / 2 , E ∣ P ( 1 / 2 + v ) ∣ 2 m ≤ ( C m / v 2 ) m + X − κ o / 2 . \mathbb{E}|U|^{2m}\leq(CmH)^m+X^{-\kappa_o/2}, \qquad\mathbb{E}|P(1/2+v)|^{2m}\leq(Cm/v^2)^m+X^{-\kappa_o/2}. E ∣ U ∣ 2 m ≤ ( C m H ) m + X − κ o /2 , E ∣ P ( 1/2 + v ) ∣ 2 m ≤ ( C m / v 2 ) m + X − κ o /2 . when m ℓ i m\ell_i m ℓ i L X L_X L X m = ⌈ κ ∗ H ⌉ m=\lceil\kappa_*H\rceil m = ⌈ κ ∗ H ⌉ A , B 0 A,B_0 A , B 0 ∣ U ∣ |U| ∣ U ∣ K 0 H K_0H K 0 H δ ∣ P ∣ \delta|P| δ ∣ P ∣ H H H O ( e − c H ) O(e^{-cH}) O ( e − cH ) log ∣ L D ( s + ) ∣ ≥ − C L H \log|L_D(s_+)|\geq-C_LH log ∣ L D ( s + ) ∣ ≥ − C L H
∣ L D ( s + ) ∣ ≤ ∣ P ℓ 1 , D ( s + ) ∣ + C e − B 0 { ∣ P ℓ 1 , D ( σ 1 ) ∣ + B L X } . |L_D(s_+)|\leq|P_{\ell_1,D}(s_+)|+Ce^{-B_0}\{|P_{\ell_1,D}(\sigma_1)|+BL_X\}. ∣ L D ( s + ) ∣ ≤ ∣ P ℓ 1 , D ( s + ) ∣ + C e − B 0 { ∣ P ℓ 1 , D ( σ 1 ) ∣ + B L X } . If Q D ≥ X b Q_D\geq X^b Q D ≥ X b B 0 B_0 B 0 B L X BL_X B L X b L X / 16 bL_X/16 b L X /16 A A A b L X / 16 bL_X/16 b L X /16
∣ L D ′ / L D ( s + ) ∣ ≤ b 4 L X . |L'_D/L_D(s_+)|\leq\frac{b}{4}L_X. ∣ L D ′ / L D ( s + ) ∣ ≤ 4 b L X . The functional equation gives the exact identities
L D ′ L D ( s − ) = − log Q D − γ D ′ γ D ( s − ) − γ D ′ γ D ( s + ) − L D ′ L D ( s + ) , \frac{L_D'}{L_D}(s_-)=-\log Q_D-\frac{\gamma_D'}{\gamma_D}(s_-)-\frac{\gamma_D'}{\gamma_D}(s_+)-\frac{L_D'}{L_D}(s_+), L D L D ′ ( s − ) = − log Q D − γ D γ D ′ ( s − ) − γ D γ D ′ ( s + ) − L D L D ′ ( s + ) , log ∣ L D ( s − ) ∣ = log ∣ L D ( s + ) ∣ + δ log Q D + O K ( δ ) . \log\lvert L_D(s_-)\rvert=\log\lvert L_D(s_+)\rvert+\delta\log Q_D+O_K(\delta). log ∣ L D ( s − )∣ = log ∣ L D ( s + )∣ + δ log Q D + O K ( δ ) . The compact gamma bounds are part of the analytic axiom. The root sign disappears from both expressions. Thus L D ′ ( s − ) ≠ 0 L_D'(s_-)\ne0 L D ′ ( s − ) = 0 log ∣ L D ′ ( s − ) ∣ ≥ − C ∗ H \log\lvert L_D'(s_-)\rvert\ge-C_*H log ∣ L D ′ ( s − )∣ ≥ − C ∗ H L D ( 1 / 2 ) ≠ 0 L_D(1/2)\ne0 L D ( 1/2 ) = 0
The central derivative second-moment axiom and Cauchy on the circle ∣ z − s − ∣ = 7 δ \lvert z-s_-\rvert=7\delta ∣ z − s − ∣ = 7 δ
E max ∣ s − s − ∣ ≤ 6 δ ∣ L D ′ ( s ) ∣ 2 ≤ 7 C A e C d ( A ) H . \mathbb{E}\max_{\lvert s-s_-\rvert\le6\delta}\lvert L_D'(s)\rvert^2\le7C_Ae^{C_d(A)H}. E ∣ s − s − ∣ ≤ 6 δ max ∣ L D ′ ( s ) ∣ 2 ≤ 7 C A e C d ( A ) H . Choose 2 K c > C d ( A ) + 1 2K_c>C_d(A)+1 2 K c > C d ( A ) + 1 3 δ 3\delta 3 δ 6 δ 6\delta 6 δ ( K c + C ∗ ) H / log 2 (K_c+C_*)H/\log2 ( K c + C ∗ ) H / log 2 O ( e − c H ) + O ( X − η c ) O(e^{-cH})+O(X^{-\eta c}) O ( e − cH ) + O ( X − η c )
Combining the lower bound, bulk upper bound and central upper bound gives the abstract theorem with exceptional probability O ( e − c H ) O(e^{-c\sqrt{H}}) O ( e − c H ) B + B_+ B + B 0 B_0 B 0 ε \varepsilon ε ε 0 \varepsilon_0 ε 0 C y C_y C y A A A X 0 X_0 X 0 A A A
On X b ≤ Q D ≤ X B X^b\le Q_D\le X^B X b ≤ Q D ≤ X B H H H log log Q D \log\log Q_D log log Q D
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