2020 Mathematics Subject Classification. Primary 11R42; Secondary 11M20, 11J81.
The research of the author is partially supported by the Simons Foundation Grant MPS-TSM-00006679.
Introduction
In this paper, we reconsider the Landau-Siegel zeros problem over arbitrary number fields.
The real-zero estimate
First, we state the real-zero estimate for arbitrary number fields and its quadratic-extension consequence. The determinant estimate of § 4 and the prime-sum comparison of § 5 supply the proof.
Let n = [ K : Q ] n = [K : \mathbb{Q}] n = [ K : Q ] D K D_K D K K K K
B K = log D K + n log ( 2 n ) + 6 n + 1. B_K = \log D_K + n \log(2n) + 6n + 1. B K = log D K + n log ( 2 n ) + 6 n + 1. All logarithms are natural.
Theorem 1.1. For every number field K K K β ∈ ( 0 , 1 ) \beta\in(0,1) β ∈ ( 0 , 1 ) ζ K ( s ) \zeta_K(s) ζ K ( s )
1 − β ≥ 1 1370 B K . (1) 1-\beta\ge\frac{1}{1370B_K}. \tag*{(1)} 1 − β ≥ 1370 B K 1 . (1) The field K K K § 5.4 ). The n log n n \log n n log n
The consequence n ≤ 2 log D K n \le2 \log D_K n ≤ 2 log D K § 8 ) gives, for K ≠ Q K \ne\mathbb{Q} K = Q
( 1 − β ) log D K ≥ 1 1370 { 14 + 2 log ( 2 n ) } ≥ 1 21920 log ( 2 n ) . (2) (1-\beta)\log D_K \ge\frac{1}{1370\{14+2\log(2n)\}} \ge\frac{1}{21920\log(2n)}. \tag*{(2)} ( 1 − β ) log D K ≥ 1370 { 14 + 2 log ( 2 n )} 1 ≥ 21920 log ( 2 n ) 1 . (2) The assertion for K = Q K = \mathbb{Q} K = Q ζ Q \zeta_{\mathbb{Q}} ζ Q ( 0 , 1 ) (0,1) ( 0 , 1 )
Corollary 1.2. Let F / K F/K F / K [ K : Q ] = n [K:\mathbb{Q}]=n [ K : Q ] = n ξ = χ F / K \xi=\chi_{F/K} ξ = χ F / K L ( β , ξ ) = 0 L(\beta,\xi)=0 L ( β , ξ ) = 0 0 < β < 1 0<\beta<1 0 < β < 1
1 − β ≥ 1 1370 { log D F + 2 n log ( 4 n ) + 12 n + 1 } . (3) 1-\beta\ge\frac{1}{1370\{\log D_F+2n\log(4n)+12n+1\}}. \tag*{(3)} 1 − β ≥ 1370 { log D F + 2 n log ( 4 n ) + 12 n + 1 } 1 . (3) Proof. The identity ζ F ( s ) = ζ K ( s ) L ( s , ξ ) \zeta_F(s)=\zeta_K(s)L(s,\xi) ζ F ( s ) = ζ K ( s ) L ( s , ξ ) β \beta β ζ F \zeta_F ζ F 2 n 2n 2 n
Unitary Hecke characters
Next, we state the uniform zero-free region for primitive unitary Hecke characters, using the normalization of § 2.1 . § 6.1 proves it from the real-zero estimate and classical positivity.
Let χ \chi χ K K K f χ \mathfrak{f}_{\chi} f χ § 2.1 , write
Λ ( s , χ ) = ( D K N f χ ) s / 2 ∏ j = 1 n Γ R ( s + μ j ) L ( s , χ ) , Γ R ( s ) = π − s / 2 Γ ( s / 2 ) . \Lambda(s,\chi) = (D_K N\mathfrak{f}_{\chi})^{s/2} \prod_{j=1}^{n} \Gamma_{\mathbb{R}}(s+\mu_j)L(s,\chi), \qquad\Gamma_{\mathbb{R}}(s) = \pi^{-s/2}\Gamma(s/2). Λ ( s , χ ) = ( D K N f χ ) s /2 j = 1 ∏ n Γ R ( s + μ j ) L ( s , χ ) , Γ R ( s ) = π − s /2 Γ ( s /2 ) . Here Γ \Gamma Γ μ j \mu_j μ j § 2.1 . Put
H ( χ , t ) = log ( D K N f χ ) + ∑ j = 1 n log ( 3 + ∣ μ j + i t ∣ ) + 4 n + 1 , (4) \mathcal{H}(\chi,t) = \log(D_K N\mathfrak{f}_{\chi}) + \sum_{j=1}^{n}\log(3+|\mu_j+it|) + 4n+1, \tag*{(4)} H ( χ , t ) = log ( D K N f χ ) + j = 1 ∑ n log ( 3 + ∣ μ j + i t ∣ ) + 4 n + 1 , (4) H + ( χ , t ) = H ( χ , t ) + n log ( 2 n ) . \mathcal{H}^{+}(\chi,t) = \mathcal{H}(\chi,t) + n\log(2n). H + ( χ , t ) = H ( χ , t ) + n log ( 2 n ) . Theorem 1.3. For every number field K K K χ \chi χ L ( s , χ ) L(s,\chi) L ( s , χ ) s = σ + i t s=\sigma+it s = σ + i t
σ > 1 − 1 4110 H + ( χ , t ) . (5) \sigma> 1-\frac{1}{4110\mathcal{H}^{+}(\chi,t)}. \tag*{(5)} σ > 1 − 4110 H + ( χ , t ) 1 . (5) Imaginary norm twists of the principal character may have their usual pole.
The constant 4110 is absolute; the degree, discriminant, conductor, infinity type and height enter through H + ( χ , t ) \mathcal{H}^{+}(\chi,t) H + ( χ , t )
The passage from real quadratic characters to arbitrary unitary characters uses classical positivity and imaginary norm twists. The half-plane theorems recalled in § 7.3 , combined with classical positivity, imply a stronger region for unitary characters; Theorem 1.3 gives a different proof with explicit dependence on the field.
Earlier arithmetic work and quantitative comparison
Next, we review the arithmetic results underlying the present argument and compare their field dependence with our bounds. § 7.2 compares the prime-sum estimates; § 7.1 justifies the normal-closure deduction.
In the first five items, χ \chi χ d d d q = ∣ d ∣ ≥ 3 q=|d|\ge3 q = ∣ d ∣ ≥ 3 0 < β < 1 0<\beta<1 0 < β < 1
OpenAI [20 ] , 1 October 2026.
L ( β , χ ) = 0 ⟹ ( 1 − β ) log q ≥ c 0 > 0 , c 0 absolute . L(\beta,\chi)=0 \quad\Longrightarrow\quad(1-\beta)\log q \ge c_0 > 0,\qquad c_0\ \text{absolute}. L ( β , χ ) = 0 ⟹ ( 1 − β ) log q ≥ c 0 > 0 , c 0 absolute . The interpolation determinant is formed in Q ( d , 2 ) \mathbb{Q}(\sqrt{d},\sqrt{2}) Q ( d , 2 ) Q ( d ) \mathbb{Q}(\sqrt{d}) Q ( d ) χ \chi χ
Banks [2 ] , 11 October 2026. An explicit exposition of OpenAI’s rational argument gives
q ≥ 10 10 , L ( β , χ ) = 0 ⟹ ( 1 − β ) log q ≥ 3 × 10 − 3 . q \ge10^{10},\qquad L(\beta,\chi)=0 \quad\Longrightarrow\quad(1-\beta)\log q \ge3\times10^{-3}. q ≥ 1 0 10 , L ( β , χ ) = 0 ⟹ ( 1 − β ) log q ≥ 3 × 1 0 − 3 . The finite computation cited there extends the bound to q ≥ 3 q\ge3 q ≥ 3
Kerr [9 ] , 15 October 2026. The pigeonhole principle and Masser’s zero estimate, in the same biquadratic field, give
L ( β , χ ) = 0 ⟹ ( 1 − β ) log q ≫ 1. L(\beta,\chi)=0 \quad\Longrightarrow\quad(1-\beta)\log q \gg1. L ( β , χ ) = 0 ⟹ ( 1 − β ) log q ≫ 1. The fixed exception d = 8 d = 8 d = 8
Green [8 ] , 17–18 October 2026. A Gram determinant of monomial derivatives in Q ( d ) \mathbb{Q}(\sqrt{d}) Q ( d ) C C C q q q
X = q C , ∑ p ≤ X χ ( p ) = − 1 log p p ≤ ( 1 2 + O ( C − 1 ) ) log X + O C ( 1 ) . X = q^{C}, \qquad\sum_{\substack{p \le X \\ \chi(p)=-1}} \frac{\log p}{p} \le\left(\frac{1}{2} + O(C^{-1})\right) \log X + O_{C}(1). X = q C , p ≤ X χ ( p ) = − 1 ∑ p log p ≤ ( 2 1 + O ( C − 1 ) ) log X + O C ( 1 ) . Barreto [3 ] , 19 October 2026. A Wronskian proof of row independence in Green’s determinant argument gives, uniformly for x ≥ 2 x \ge2 x ≥ 2
∑ p ≤ x χ ( p ) = − 1 log p p ≤ 1 2 log x + 1 4 log q + log log ( 3 x ) + O ( 1 ) , \sum_{\substack{p \le x \\ \chi(p)=-1}} \frac{\log p}{p} \le\frac{1}{2}\log x + \frac{1}{4}\log q + \log\log(3x) + O(1), p ≤ x χ ( p ) = − 1 ∑ p log p ≤ 2 1 log x + 4 1 log q + log log ( 3 x ) + O ( 1 ) , and consequently
L ( β , χ ) = 0 ⟹ ( 1 − β ) log q ≥ 1 − O ( log log ( 3 q ) log q ) . L(\beta,\chi)=0 \quad\Longrightarrow\quad(1-\beta)\log q \ge1-O\left(\frac{\log\log(3q)}{\log q}\right). L ( β , χ ) = 0 ⟹ ( 1 − β ) log q ≥ 1 − O ( log q log log ( 3 q ) ) . Kominers [10 ] , 22 October 2026. Let L / Q L/\mathbb{Q} L / Q C ⊂ Gal ( L / Q ) C \subset\operatorname{Gal}(L/\mathbb{Q}) C ⊂ Gal ( L / Q ) c c c L L L 2 X 2X 2 X H ≥ 1 H \ge1 H ≥ 1
∑ p ≤ X Frob p = C log p p ≤ c log H + log log ( 3 X ) + O ( c ) , X ≥ 2 , \sum_{\substack{p \le X \\ \operatorname{Frob}_{p}=C}} \frac{\log p}{p} \le c\log H+\log\log(3X)+O(c), \qquad X \ge2, p ≤ X Frob p = C ∑ p log p ≤ c log H + log log ( 3 X ) + O ( c ) , X ≥ 2 , with an absolute implied constant; ramified primes are omitted. This yields weighted Chebotarev for fixed L L L
Banks [1 ] , 22 October 2026. For a number field E E E ∅ ≠ S ⊂ Hom ( E , C ) \varnothing\ne S \subset\operatorname{Hom}(E,\mathbb{C}) ∅ = S ⊂ Hom ( E , C )
Γ = { ( σ θ ) σ ∈ S : θ ∈ O E } , \Gamma=\{(\sigma\theta)_{\sigma\in S}:\theta\in\mathcal{O}_{E}\}, Γ = {( σ θ ) σ ∈ S : θ ∈ O E } , μ ( Γ ) = [ E : Q ] ∣ S ∣ \mu(\Gamma)=\frac{[E:\mathbb{Q}]}{|S|} μ ( Γ ) = ∣ S ∣ [ E : Q ] [1 ] , Proposition 4.2 and its proof.
Here μ ( Γ ) \mu(\Gamma) μ ( Γ ) Theorem 4.1 ; Hom ( E , C ) \operatorname{Hom}(E,\mathbb{C}) Hom ( E , C ) L / k L/k L / k c B ( k ) c_{B}(k) c B ( k ) k k k k k k
In the present argument, choose the smallest rational prime a a a a ∉ K \sqrt{a}\notin K a ∈ / K E = K ( a ) E=K(\sqrt{a}) E = K ( a ) d = 2 n d=2n d = 2 n s = d − 1 s=d-1 s = d − 1 ι \iota ι K K K E E E ι \iota ι S S S H 0 ≥ 2 H_{0}\ge2 H 0 ≥ 2 D ≥ 2 H 0 D\ge2H_{0} D ≥ 2 H 0
log λ E ≤ 2 d log D E , W S 1 ≤ 1 + s − 1 H 0 , \log\lambda_{E}\le\frac{2}{d}\log D_{E}, \qquad\frac{W}{S_{1}}\le1+\frac{s-1}{H_{0}}, log λ E ≤ d 2 log D E , S 1 W ≤ 1 + H 0 s − 1 , p ∤ D E ⟹ 1 − δ p ≤ r K ( p ) n . p\nmid D_{E}\quad\Longrightarrow\quad1-\delta_{p}\le\frac{r_{K}(p)}{n}. p ∤ D E ⟹ 1 − δ p ≤ n r K ( p ) . The notation is defined in § 4 –§ 5 . The real-zero bias, ramification bound, and cutoff in § 5 , with cancellation of the normal-closure degree, yield
K arbitrary , ζ K ( β ) = 0 ⟹ ( 1 − β ) { log D K + n log ( 2 n ) + 6 n + 1 } ≥ 1 1370 . K\text{ arbitrary}, \quad\zeta_{K}(\beta)=0 \quad\Longrightarrow\quad(1-\beta)\{\log D_{K}+n\log(2n)+6n+1\}\ge\frac{1}{1370}. K arbitrary , ζ K ( β ) = 0 ⟹ ( 1 − β ) { log D K + n log ( 2 n ) + 6 n + 1 } ≥ 1370 1 . Positivity and norm twists give (5) . The normal-closure comparison below is proved in § 7 .
Banks’ Corollary 1.2 [1 ] states that for each number field K K K c B ( K ) > 0 c_{\mathrm{B}}(K)>0 c B ( K ) > 0
( 1 − β ) log D F ≥ c B ( K ) ( F / K quadratic , L ( β , χ F / K ) = 0 , 0 < β < 1 ) . (6) (1-\beta)\log D_F\ge c_{\mathrm{B}}(K)\qquad(F/K\text{ quadratic},\ L(\beta,\chi_{F/K})=0,\ 0<\beta<1). \tag*{(6)} ( 1 − β ) log D F ≥ c B ( K ) ( F / K quadratic , L ( β , χ F / K ) = 0 , 0 < β < 1 ) . (6) Corollary 1.2 gives a fully displayed alternative. For K ≠ Q K\ne\mathbb{Q} K = Q D F = D K 2 N f ξ D_F=D_K^2N\mathfrak{f}_\xi D F = D K 2 N f ξ (6) may be replaced by
c exp ( K ) = 1 1370 ( 1 + 2 n log ( 4 n ) + 12 n + 1 2 log D K ) . (7) c_{\exp}(K)=\frac{1}{1370\left(1+\frac{2n\log(4n)+12n+1}{2\log D_K}\right)}. \tag*{(7)} c e x p ( K ) = 1370 ( 1 + 2 l o g D K 2 n l o g ( 4 n ) + 12 n + 1 ) 1 . (7) Indeed, putting C n = 2 n log ( 4 n ) + 12 n + 1 C_n=2n\log(4n)+12n+1 C n = 2 n log ( 4 n ) + 12 n + 1
( 1 − β ) log D F ≥ 1 1370 ( 1 + C n / log D F ) ≥ c exp ( K ) . (1-\beta)\log D_F\ge\frac{1}{1370(1+C_n/\log D_F)}\ge c_{\exp}(K). ( 1 − β ) log D F ≥ 1370 ( 1 + C n / log D F ) 1 ≥ c e x p ( K ) . This is an admissible explicit constant, not an upper or lower estimate for Banks’ particular c B ( K ) c_{\mathrm{B}}(K) c B ( K )
Banks’ Theorem 1.3 [1 ] , with base field Q \mathbb{Q} Q c B > 0 c_{\mathrm{B}}>0 c B > 0
( 1 − β ) log D L ≥ c B ( L / Q Galois , ζ L ( β ) = 0 , 0 < β < 1 ) . (8) (1-\beta)\log D_L\ge c_{\mathrm{B}}\qquad(L/\mathbb{Q}\text{ Galois},\ \zeta_L(\beta)=0,\ 0<\beta<1). \tag*{(8)} ( 1 − β ) log D L ≥ c B ( L / Q Galois , ζ L ( β ) = 0 , 0 < β < 1 ) . (8) Banks’ Remark 3.4 also treats almost-normal fields. For an arbitrary degree-n n n K K K
( 1 − β ) log D K ≥ 2 c B n ! . (9) (1-\beta)\log D_K\ge\frac{2c_{\mathrm{B}}}{n!}. \tag*{(9)} ( 1 − β ) log D K ≥ n ! 2 c B . (9) Thus (9) loses n ! n! n ! (2) loses only log ( 2 n ) \log(2n) log ( 2 n ) [1 ] ; no pointwise comparison with its fixed-base constant is asserted. Deductions from [1 ] , Theorem 1.5 and Stark-type reductions have not been optimized here. For the degree-ten field of Example 7.2, with the admissible constant c B ≤ 1 / 4 c_{\mathrm{B}}\le1/4 c B ≤ 1/4 Theorem 1.1 exceeds the width from (9) by a factor greater than 10 71 10^{71} 1 0 71
The lower comparison uses rational primes and degree-one primes of K K K [1 ] , § 3.2 .
The arithmetic argument
Finally, we outline the determinant construction of § 4 and the real-zero comparison of § 5 . Appendix § 8 collects the numerical inequalities.
Choose a rational prime a a a a ∉ K \sqrt{a}\notin K a ∈ / K E = K ( a ) E=K(\sqrt{a}) E = K ( a ) d = 2 n d=2n d = 2 n
2 log D K ≤ log D E ≤ 4 log D K + n log 8. 2\log D_K\le\log D_E\le4\log D_K+n\log8. 2 log D K ≤ log D E ≤ 4 log D K + n log 8. Use all embeddings of E E E K K K [1 ] , Proposition 4.2 and its proof give a nonzero evaluation determinant. Its points lie in E E E M M M [ M : Q ] [M:\mathbb{Q}] [ M : Q ]
∑ p ≤ X δ p log p p ≤ ( 1 − 15 16 d ) log X + 4 d log D E + 32 log ( 2 d ) ( X ∈ Z , X ≥ d 2 ) . \sum_{p\le X}\delta_p\frac{\log p}{p}\le\left(1-\frac{15}{16d}\right)\log X+\frac{4}{d}\log D_E+32\log(2d)\qquad(X\in\mathbb{Z},\ X\ge d^2). p ≤ X ∑ δ p p log p ≤ ( 1 − 16 d 15 ) log X + d 4 log D E + 32 log ( 2 d ) ( X ∈ Z , X ≥ d 2 ) . The omitted embedding relates 1 − δ p 1-\delta_p 1 − δ p K K K β ∈ ( 0 , 1 ) \beta\in(0,1) β ∈ ( 0 , 1 ) ζ K \zeta_K ζ K Δ = 1 − β \Delta=1-\beta Δ = 1 − β H = H K + ( 1 , 0 ) H=\mathcal{H}_K^+(1,0) H = H K + ( 1 , 0 ) 1 1 1 K K K
− ζ K ′ ζ K ( 1 + h ) ≤ Δ h ( h + Δ ) + 1 + 1 2 H K ( 1 , 0 ) ( 0 < h ≤ 1 ) . -\frac{\zeta_K'}{\zeta_K}(1+h) \le\frac{\Delta}{h(h+\Delta)}+1+\frac{1}{2}\mathcal{H}_K(1,0) \qquad(0<h\le1). − ζ K ζ K ′ ( 1 + h ) ≤ h ( h + Δ ) Δ + 1 + 2 1 H K ( 1 , 0 ) ( 0 < h ≤ 1 ) . The prime-sum comparison and ramification bound (§ 5.3 ), with
X = 2 ⌈ 176 H / log 2 ⌉ , h = 2 log X , X=2^{\lceil176H/\log2\rceil}, \qquad h=\frac{2}{\log X}, X = 2 ⌈ 176 H / l o g 2 ⌉ , h = log X 2 , give Δ H ≥ 1 / 1370 \Delta H\ge1/1370 Δ H ≥ 1/1370 H ≤ B K H\le B_K H ≤ B K 3 ⋅ 1370 = 4110 3\cdot1370=4110 3 ⋅ 1370 = 4110
The arithmetic results used here are recalled in § 1.3 ; the small-integer estimate is [6 Proposition 1] . The quantitative refinement combines evaluation points in E E E [20 , 2 ] nor a zero-free half-plane theorem is used.
The half-plane results of [7 , 12 , 13 , 14 ] give stronger nonvanishing conclusions (§ 7.3 ). The contribution here is the alternative arithmetic proof with explicit field dependence. For the earlier fixed-field refinement over Q ( − 3 ) \mathbb{Q}(\sqrt{-3}) Q ( − 3 ) [11 ] .
Here is the structure of the paper. § 2 fixes notation and records rational-prime estimates. § 3 gives the gamma-factor, logarithmic-derivative and positivity estimates. § 4 develops the embedding determinant estimate. § 5 proves the real-zero bound, including the auxiliary quadratic extension, ramification bound and cutoff choice. § 6 treats unitary Hecke characters, analytic conductors, imprimitive characters and norm twists. § 7 compares the arithmetic estimates with Banks’ results and recalls earlier half-plane theorems. Appendix § 8 collects the numerical inequalities.
Acknowledgments
OpenAI Codex (GPT 6.1 Sol Medium-Ultra) and Anthropic Claude (Opus 5.5 Medium) assisted with drafting, reasoning, checking references, and auditing. The Lean formalization is in progress, based on OpenAI’s accompanying Lean sources [22 ] .
Notation and preliminaries
In this section, we fix the arithmetic notation, Hecke normalizations, and elementary prime estimates. § 3 uses the Hecke heights, while § 4 –§ 5 use the house bounds and rational-prime estimates.
For a number field E E E O E \mathcal{O}_E O E D E D_E D E ζ E \zeta_E ζ E a ⊂ O E \mathfrak{a}\subset\mathcal{O}_E a ⊂ O E N a = ∣ O E / a ∣ N\mathfrak{a}=\lvert\mathcal{O}_E/\mathfrak{a}\rvert N a = ∣ O E / a ∣ N E / Q N_{E/\mathbb{Q}} N E / Q θ ∈ E \theta\in E θ ∈ E max σ : E ↪ C ∣ σ θ ∣ \max_{\sigma:E\hookrightarrow\mathbb{C}}\lvert\sigma\theta\rvert max σ : E ↪ C ∣ σ θ ∣ Q \mathbb{Q} Q
The letter p p p p \mathfrak{p} p P P P p \mathfrak{p} p Frob P \operatorname{Frob}_P Frob P x ↦ x N p x\mapsto x^{N\mathfrak{p}} x ↦ x N p Frob p \operatorname{Frob}_{\mathfrak{p}} Frob p 1 P \mathbf{1}_{\mathcal{P}} 1 P P \mathcal{P} P v p ( m ) v_p(m) v p ( m ) p p p m m m O O O ≪ \ll ≪ ≫ \gg ≫ ≍ \asymp ≍
Conventions. In Sections 4–5 the letter s s s D D D H 0 H_0 H 0 E p E_p E p D K D_K D K D E D_E D E H H H H ( χ , t ) \mathcal{H}(\chi,t) H ( χ , t )
Hecke characters and heights
In this subsection, we fix the gamma factors, norm-twist convention, and logarithmic heights used in Section 3. These normalizations also govern the zero-free regions of Section 6.
Let χ \chi χ f χ \mathfrak{f}_\chi f χ A χ = D K N f χ A_\chi=D_KN\mathfrak{f}_\chi A χ = D K N f χ L L L
L ( s , χ ) = ∏ p ( 1 − χ ( p ) ( N p ) − s ) − 1 ( Re s > 1 ) . L(s,\chi)=\prod_{\mathfrak{p}}(1-\chi(\mathfrak{p})(N\mathfrak{p})^{-s})^{-1}\qquad(\operatorname{Re}s>1). L ( s , χ ) = p ∏ ( 1 − χ ( p ) ( N p ) − s ) − 1 ( Re s > 1 ) . where χ ( p ) = 0 \chi(\mathfrak{p})=0 χ ( p ) = 0 p ∣ f χ \mathfrak{p}\mid\mathfrak{f}_\chi p ∣ f χ ∣ χ ( p ) ∣ = 1 |\chi(\mathfrak{p})|=1 ∣ χ ( p ) ∣ = 1 1 1 1 L ( s , 1 ) = ζ K ( s ) L(s,1)=\zeta_K(s) L ( s , 1 ) = ζ K ( s )
χ τ ( a ) = χ ( a ) ( N a ) − i τ , L ( s , χ τ ) = L ( s + i τ , χ ) . (10) \chi_\tau(\mathfrak{a})=\chi(\mathfrak{a})(N\mathfrak{a})^{-i\tau},\qquad L(s,\chi_\tau)=L(s+i\tau,\chi). \tag*{(10)} χ τ ( a ) = χ ( a ) ( N a ) − i τ , L ( s , χ τ ) = L ( s + i τ , χ ) . (10) This convention fixes signs independently of idele absolute-value conventions. Recall Euler’s gamma function
Γ ( z ) = ∫ 0 ∞ e − x x z − 1 d x ( Re z > 0 ) , \Gamma(z)=\int_0^\infty e^{-x}x^{z-1}\,dx\qquad(\operatorname{Re}z>0), Γ ( z ) = ∫ 0 ∞ e − x x z − 1 d x ( Re z > 0 ) , with its meromorphic continuation. We write the completed function in real gamma factors as
Λ ( s , χ ) = A χ s / 2 ∏ j = 1 n Γ R ( s + μ j ) L ( s , χ ) , Γ R ( z ) = π − z / 2 Γ ( z / 2 ) . (11) \Lambda(s,\chi)=A_\chi^{s/2}\prod_{j=1}^{n}\Gamma_{\mathbb{R}}(s+\mu_j)L(s,\chi),\qquad\Gamma_{\mathbb{R}}(z)=\pi^{-z/2}\Gamma(z/2). \tag*{(11)} Λ ( s , χ ) = A χ s /2 j = 1 ∏ n Γ R ( s + μ j ) L ( s , χ ) , Γ R ( z ) = π − z /2 Γ ( z /2 ) . (11) Normalizing constants independent of s s s ϵ + i τ v \epsilon+i\tau_v ϵ + i τ v ϵ ∈ { 0 , 1 } \epsilon\in\{0,1\} ϵ ∈ { 0 , 1 } ∣ k v ∣ / 2 + i τ v |k_v|/2+i\tau_v ∣ k v ∣/2 + i τ v ∣ k v ∣ / 2 + 1 + i τ v |k_v|/2+1+i\tau_v ∣ k v ∣/2 + 1 + i τ v τ v ∈ R \tau_v\in\mathbb{R} τ v ∈ R k v ∈ Z k_v\in\mathbb{Z} k v ∈ Z j = 1 , … , n j=1,\ldots,n j = 1 , … , n Re μ j ≥ 0 \operatorname{Re}\mu_j\ge0 Re μ j ≥ 0
Definition 2.1 (Explicit logarithmic height). For t ∈ R t\in\mathbb{R} t ∈ R
H ( χ , t ) = log A χ + ∑ j = 1 n log ( 3 + ∣ μ j + i t ∣ ) + 4 n + 1. (12) \mathcal{H}(\chi,t)=\log A_\chi+\sum_{j=1}^{n}\log(3+|\mu_j+it|)+4n+1. \tag*{(12)} H ( χ , t ) = log A χ + j = 1 ∑ n log ( 3 + ∣ μ j + i t ∣ ) + 4 n + 1. (12) We will also use the height including the degree term
H + ( χ , t ) = H ( χ , t ) + n log ( 2 n ) . (13) \mathcal{H}^{+}(\chi,t)=\mathcal{H}(\chi,t)+n\log(2n). \tag*{(13)} H + ( χ , t ) = H ( χ , t ) + n log ( 2 n ) . (13) For the principal character, L ( s , 1 ) = ζ K ( s ) L(s,1)=\zeta_K(s) L ( s , 1 ) = ζ K ( s ) O K \mathcal{O}_K O K μ j ∈ { 0 , 1 } \mu_j\in\{0,1\} μ j ∈ { 0 , 1 }
An imaginary norm twist changes every parameter by i τ i\tau i τ H ( χ τ , t ) = H ( χ , t + τ ) \mathcal{H}(\chi_\tau,t)=\mathcal{H}(\chi,t+\tau) H ( χ τ , t ) = H ( χ , t + τ ) 1 τ 1_\tau 1 τ H E \mathcal{H}_E H E H E + \mathcal{H}^{+}_E H E + s = σ + i t s=\sigma+it s = σ + i t σ \sigma σ t t t
Rational-prime estimates
In this subsection, we record the rational-prime estimates used in § 5.1 and § 5.3 . Their explicit constants keep the subsequent comparison uniform in the field.
For x ≥ 2 x \ge2 x ≥ 2
ϑ ( x ) = ∑ p ≤ x log p ≤ 4 x , ∑ p ≤ x log p p ≥ log x − 6. (14) \vartheta(x)=\sum_{p\le x}\log p\le4x,\qquad\sum_{p\le x}\frac{\log p}{p}\ge\log x-6. \tag*{(14)} ϑ ( x ) = p ≤ x ∑ log p ≤ 4 x , p ≤ x ∑ p log p ≥ log x − 6. (14) For completeness, the binomial coefficients ( 2 k k ) \binom{2k}{k} ( k 2 k ) ( 2 k + 1 k ) \binom{2k+1}{k} ( k 2 k + 1 ) N = ⌊ x ⌋ N=\lfloor x\rfloor N = ⌊ x ⌋ log N ! ≥ N log N − N \log N!\ge N\log N-N log N ! ≥ N log N − N
log N ! ≤ N ∑ p j ≤ N log p p j . \log N!\le N\sum_{p^j\le N}\frac{\log p}{p^j}. log N ! ≤ N p j ≤ N ∑ p j log p . The contribution from j ≥ 2 j\ge2 j ≥ 2 ∑ k ≥ 2 log k / ( k ( k − 1 ) ) < 4 \sum_{k\ge2}\log k/(k(k-1))<4 ∑ k ≥ 2 log k / ( k ( k − 1 )) < 4 log ( j + 1 ) ≤ log 2 + log j \log(j+1)\le\log2+\log j log ( j + 1 ) ≤ log 2 + log j ∑ j ≥ 2 j − 2 < 2 \sum_{j\ge2}j^{-2}<2 ∑ j ≥ 2 j − 2 < 2 ∑ j ≥ 2 ( log j ) / j 2 < 2 \sum_{j\ge2}(\log j)/j^2<2 ∑ j ≥ 2 ( log j ) / j 2 < 2 log ⌊ x ⌋ ≥ log x − log 2 \log\lfloor x\rfloor\ge\log x-\log2 log ⌊ x ⌋ ≥ log x − log 2
Archimedean estimates and classical positivity
In this section, we establish logarithmic-derivative and positivity estimates using the heights of § 2.1 . Together with the real-zero bound of § 5 , they yield the Hecke theorem in § 6.1 . All displayed numerical constants are independent of K K K
The analytic inputs are Hecke’s continuation and functional equation, with the principal norm twists as the only pole cases [24 Theorems 5.16 and 5.22] , and the order-one property [25 Section 1, pp. 281–282] . To pass from the normalized characters in the latter source to our convention, put
τ 0 = 1 n ∑ j = 1 n Im μ j , χ 0 = χ − τ 0 , ∑ j Im μ j ( χ 0 ) = 0. \tau_0=\frac{1}{n}\sum_{j=1}^{n}\operatorname{Im}\mu_j,\qquad\chi^0=\chi_{-\tau_0},\qquad\sum_j\operatorname{Im}\mu_j(\chi^0)=0. τ 0 = n 1 j = 1 ∑ n Im μ j , χ 0 = χ − τ 0 , j ∑ Im μ j ( χ 0 ) = 0. The conductor is unchanged, and
Λ ( s , χ ) = A χ − i τ 0 / 2 Λ ( s + i τ 0 , χ 0 ) . \Lambda(s,\chi)=A_\chi^{-i\tau_0/2}\Lambda(s+i\tau_0,\chi^0). Λ ( s , χ ) = A χ − i τ 0 /2 Λ ( s + i τ 0 , χ 0 ) . Thus the function
Ξ χ ( s ) = { ( s + i τ ) ( s + i τ − 1 ) Λ ( s , χ ) , χ = 1 τ , Λ ( s , χ ) , χ ≠ 1 τ for every τ ∈ R , \Xi_\chi(s)=
\begin{cases}
(s+i\tau)(s+i\tau-1)\Lambda(s,\chi), & \chi=1_\tau,\\
\Lambda(s,\chi), & \chi\ne1_\tau\text{ for every }\tau\in\mathbb{R},
\end{cases} Ξ χ ( s ) = { ( s + i τ ) ( s + i τ − 1 ) Λ ( s , χ ) , Λ ( s , χ ) , χ = 1 τ , χ = 1 τ for every τ ∈ R , is entire of order one and satisfies Ξ χ ( s ) = w χ Ξ χ ( 1 − s ) \Xi_\chi(s)=w_\chi\Xi_\chi(1-s) Ξ χ ( s ) = w χ Ξ χ ( 1 − s ) ∣ w χ ∣ = 1 |w_\chi|=1 ∣ w χ ∣ = 1 [25 Section 1] ; we do not use its zero-free region.
Boundary nonvanishing follows directly from Euler-product positivity. Let χ [ 2 ] \chi^{[2]} χ [ 2 ] χ 2 \chi^2 χ 2 L ( 1 + i t , χ ) = 0 L(1+it,\chi)=0 L ( 1 + i t , χ ) = 0 m ≥ 1 m\ge1 m ≥ 1 χ [ 2 ] ≠ 1 − 2 t \chi^{[2]}\ne1_{-2t} χ [ 2 ] = 1 − 2 t h ↓ 0 h\downarrow0 h ↓ 0
0 ≤ − 3 ζ K ′ ζ K ( 1 + h ) − 4 Re L ′ L ( 1 + h + i t , χ ) − Re L ′ L ( 1 + h + 2 i t , χ [ 2 ] ) ≤ 3 − 4 m h + O χ , t ( 1 ) , \begin{aligned}
0\le-3\frac{\zeta'_K}{\zeta_K}(1+h)-4\operatorname{Re}\frac{L'}{L}(1+h+it,\chi)-\operatorname{Re}\frac{L'}{L}(1+h+2it,\chi^{[2]}) \\
&\le\frac{3-4m}{h}+O_{\chi,t}(1),
\end{aligned} 0 ≤ − 3 ζ K ζ K ′ ( 1 + h ) − 4 Re L L ′ ( 1 + h + i t , χ ) − Re L L ′ ( 1 + h + 2 i t , χ [ 2 ] ) ≤ h 3 − 4 m + O χ , t ( 1 ) , a contradiction. At a prime ramified for χ \chi χ 3 − 1 = 2 3-1=2 3 − 1 = 2 ξ = χ t \xi=\chi_t ξ = χ t ξ 2 = 1 \xi^2=1 ξ 2 = 1 L ( 1 + i t , χ ) = L ( 1 , ξ ) L(1+it,\chi)=L(1,\xi) L ( 1 + i t , χ ) = L ( 1 , ξ ) ξ = 1 \xi=1 ξ = 1 F / K F/K F / K ξ \xi ξ
L ( 1 , ξ ) = Res s = 1 ζ F ( s ) Res s = 1 ζ K ( s ) > 0. L(1,\xi)=\frac{\operatorname{Res}_{s=1}\zeta_F(s)}{\operatorname{Res}_{s=1}\zeta_K(s)}>0. L ( 1 , ξ ) = Res s = 1 ζ K ( s ) Res s = 1 ζ F ( s ) > 0. Together with the functional equation and the Euler product, this puts all zeros of Ξ χ \Xi_\chi Ξ χ 0 < Re s < 1 0<\operatorname{Re}s<1 0 < Re s < 1
Gamma factors and logarithmic derivatives
In this subsection, we bound the gamma contributions and express logarithmic derivatives in terms of zeros, with the normalization of § 2.1 . § 3.2 uses these estimates to isolate possible real zeros.
Lemma 3.1 (Gamma bound). For Re z ≥ 1 / 2 \operatorname{Re}z\ge1/2 Re z ≥ 1/2
Re Γ ′ Γ ( z ) ≤ log ( 1 + ∣ z ∣ ) + 4. \operatorname{Re}\frac{\Gamma'}{\Gamma}(z)\le\log(1+|z|)+4. Re Γ Γ ′ ( z ) ≤ log ( 1 + ∣ z ∣ ) + 4. Consequently, for 1 < σ ≤ 2 1<\sigma\le2 1 < σ ≤ 2
1 2 log A χ + ∑ j Re Γ ′ Γ ( σ + i t + μ j ) ≤ 1 2 H ( χ , t ) . (15) \frac{1}{2}\log A_\chi+\sum_j\operatorname{Re}\frac{\Gamma'}{\Gamma}(\sigma+it+\mu_j)\le\frac{1}{2}\mathcal{H}(\chi,t). \tag*{(15)} 2 1 log A χ + j ∑ Re Γ Γ ′ ( σ + i t + μ j ) ≤ 2 1 H ( χ , t ) . (15) Proof. Write z = x + i y z=x+iy z = x + i y r = ∣ z ∣ r=|z| r = ∣ z ∣ N = ⌈ r ⌉ ≥ 1 N=\lceil r\rceil\ge1 N = ⌈ r ⌉ ≥ 1 H N = ∑ j = 1 N j − 1 H_N=\sum_{j=1}^{N}j^{-1} H N = ∑ j = 1 N j − 1 γ = lim N → ∞ ( H N − log N ) \gamma=\lim_{N\to\infty}(H_N-\log N) γ = lim N → ∞ ( H N − log N )
Γ ′ Γ ( z ) = − γ + ∑ k ≥ 0 ( 1 k + 1 − 1 k + z ) , \frac{\Gamma'}{\Gamma}(z)=-\gamma+\sum_{k\ge0}\left(\frac{1}{k+1}-\frac{1}{k+z}\right), Γ Γ ′ ( z ) = − γ + k ≥ 0 ∑ ( k + 1 1 − k + z 1 ) , the first N N N
∑ k = 0 N − 1 ( 1 k + 1 − Re 1 k + z ) ≤ H N . \sum_{k=0}^{N-1}\left(\frac{1}{k+1}-\operatorname{Re}\frac{1}{k+z}\right)\le H_N. k = 0 ∑ N − 1 ( k + 1 1 − Re k + z 1 ) ≤ H N . For k ≥ N k\ge N k ≥ N
1 k + 1 − k + x ( k + x ) 2 + y 2 = x − 1 ( k + 1 ) ( k + x ) + y 2 ( k + x ) ( ( k + x ) 2 + y 2 ) . \frac{1}{k+1}-\frac{k+x}{(k+x)^2+y^2}=\frac{x-1}{(k+1)(k+x)}+\frac{y^2}{(k+x)((k+x)^2+y^2)}. k + 1 1 − ( k + x ) 2 + y 2 k + x = ( k + 1 ) ( k + x ) x − 1 + ( k + x ) (( k + x ) 2 + y 2 ) y 2 . The sum of the positive parts of the first terms is at most 1 1 1 x > 1 x>1 x > 1 ( N − 1 ) ∑ k ≥ N k − 2 ≤ 1 − N − 2 (N-1)\sum_{k\ge N}k^{-2}\le1-N^{-2} ( N − 1 ) ∑ k ≥ N k − 2 ≤ 1 − N − 2 r 2 ( N − 3 + ( 2 N 2 ) − 1 ) ≤ 3 / 2 r^2(N^{-3}+(2N^2)^{-1})\le3/2 r 2 ( N − 3 + ( 2 N 2 ) − 1 ) ≤ 3/2 H N − γ ≤ log N + 1 H_N-\gamma\le\log N+1 H N − γ ≤ log N + 1 z = ( σ + i t + μ j ) / 2 z=(\sigma+it+\mu_j)/2 z = ( σ + i t + μ j ) /2 1 + ∣ z ∣ ≤ 3 + ∣ μ j + i t ∣ 1+|z|\le3+|\mu_j+it| 1 + ∣ z ∣ ≤ 3 + ∣ μ j + i t ∣ − 1 2 log π -\frac{1}{2}\log\pi − 2 1 log π
Lemma 3.2 (Logarithmic derivative). If χ \chi χ , for 1 < σ ≤ 2 1<\sigma\le2 1 < σ ≤ 2
− Re L ′ L ( σ + i t , χ ) ≤ 1 2 H ( χ , t ) − ∑ ρ Re 1 σ + i t − ρ . (16) -\operatorname{Re}\frac{L'}{L}(\sigma+it,\chi)\le\frac{1}{2}\mathcal{H}(\chi,t)-\sum_\rho\operatorname{Re}\frac{1}{\sigma+it-\rho}. \tag*{(16)} − Re L L ′ ( σ + i t , χ ) ≤ 2 1 H ( χ , t ) − ρ ∑ Re σ + i t − ρ 1 . (16) The sum is over nontrivial zeros ρ \rho ρ L ( s , χ ) L(s,\chi) L ( s , χ ) 0 < Re ρ < 1 0 < \operatorname{Re}\rho< 1 0 < Re ρ < 1 χ = 1 τ \chi= 1_\tau χ = 1 τ
Re ( 1 σ + i ( t + τ ) − 1 + 1 σ + i ( t + τ ) ) (17) \operatorname{Re}\left(\frac{1}{\sigma+i(t+\tau)-1}+\frac{1}{\sigma+i(t+\tau)}\right) \tag*{(17)} Re ( σ + i ( t + τ ) − 1 1 + σ + i ( t + τ ) 1 ) (17) to the right side.
Proof. Apply Hadamard factorization to the entire function Ξ χ \Xi_\chi Ξ χ
Ξ χ ( s ) = e a + b s ∏ ρ ( 1 − s / ρ ) e s / ρ , Ξ χ ′ Ξ χ ( s ) = b + ∑ ρ ( 1 s − ρ + 1 ρ ) . \Xi_\chi(s)=e^{a+bs}\prod_\rho(1-s/\rho)e^{s/\rho},\qquad\frac{\Xi_\chi'}{\Xi_\chi}(s)=b+\sum_\rho\left(\frac{1}{s-\rho}+\frac{1}{\rho}\right). Ξ χ ( s ) = e a + b s ρ ∏ ( 1 − s / ρ ) e s / ρ , Ξ χ Ξ χ ′ ( s ) = b + ρ ∑ ( s − ρ 1 + ρ 1 ) . Its order implies ∑ ρ ∣ ρ ∣ − 2 < ∞ \sum_\rho|\rho|^{-2}<\infty ∑ ρ ∣ ρ ∣ − 2 < ∞ 0 < Re ρ < 1 0 < \operatorname{Re}\rho< 1 0 < Re ρ < 1 s 0 = 1 / 2 + i v s_0=1/2+iv s 0 = 1/2 + i v Ξ χ ( s 0 ) ≠ 0 \Xi_\chi(s_0)\ne0 Ξ χ ( s 0 ) = 0 Re ( Ξ χ ′ / Ξ χ ) ( s 0 ) = 0 \operatorname{Re}(\Xi_\chi'/\Xi_\chi)(s_0)=0 Re ( Ξ χ ′ / Ξ χ ) ( s 0 ) = 0 ρ ↔ 1 − ρ ‾ \rho\leftrightarrow1-\overline{\rho} ρ ↔ 1 − ρ
∑ ρ Re 1 s 0 − ρ = 0 , Re b = − ∑ ρ Re 1 ρ . \sum_\rho\operatorname{Re}\frac{1}{s_0-\rho}=0,\qquad\operatorname{Re}b=-\sum_\rho\operatorname{Re}\frac{1}{\rho}. ρ ∑ Re s 0 − ρ 1 = 0 , Re b = − ρ ∑ Re ρ 1 . Zeros on Re ρ = 1 / 2 \operatorname{Re}\rho=1/2 Re ρ = 1/2 s 0 s_0 s 0 v v v
Re Ξ χ ′ Ξ χ ( s ) = ∑ ρ Re 1 s − ρ . \operatorname{Re}\frac{\Xi_\chi'}{\Xi_\chi}(s)=\sum_\rho\operatorname{Re}\frac{1}{s-\rho}. Re Ξ χ Ξ χ ′ ( s ) = ρ ∑ Re s − ρ 1 . Insert the definition of Ξ χ \Xi_\chi Ξ χ Lemma 3.1 . For χ = 1 τ \chi=1_\tau χ = 1 τ (17) .
Classical positivity
In this subsection, we apply the logarithmic-derivative bounds of § 3.1 to classical positive trigonometric sums. The resulting estimates, combined with § 5 , prove Theorem 1.3 in § 6.1 .
The finite conductor of χ [ 2 ] \chi^{[2]} χ [ 2 ] f χ f_\chi f χ μ j [ 2 ] \mu_j^{[2]} μ j [ 2 ] χ \chi χ
H ( 1 , 0 ) ≤ H ( χ , t ) , H ( χ [ 2 ] , 2 t ) ≤ 2 H ( χ , t ) . (18) \mathcal{H}(1,0)\le\mathcal{H}(\chi,t),\qquad\mathcal{H}(\chi^{[2]},2t)\le2\mathcal{H}(\chi,t). \tag*{(18)} H ( 1 , 0 ) ≤ H ( χ , t ) , H ( χ [ 2 ] , 2 t ) ≤ 2 H ( χ , t ) . (18) For the first inequality, ∣ μ j + i t ∣ ≥ Re μ j |\mu_j+it|\ge\operatorname{Re}\mu_j ∣ μ j + i t ∣ ≥ Re μ j 3 + ∣ μ j [ 2 ] + 2 i t ∣ ≤ 2 ( 3 + ∣ μ j + i t ∣ ) 3+|\mu_j^{[2]}+2it|\le2(3+|\mu_j+it|) 3 + ∣ μ j [ 2 ] + 2 i t ∣ ≤ 2 ( 3 + ∣ μ j + i t ∣ ) n log 2 n\log2 n log 2 4 n + 1 4n+1 4 n + 1
Proposition 3.3 (Characters not equivalent to real ones). If χ \chi χ β + i t \beta+it β + i t
( 1 − β ) H ( χ , t ) ≥ 1 / 112. (19) (1-\beta)\mathcal{H}(\chi,t)\ge1/112. \tag*{(19)} ( 1 − β ) H ( χ , t ) ≥ 1/112. (19) Proof. Put H = H ( χ , t ) H=\mathcal{H}(\chi,t) H = H ( χ , t ) χ \chi χ χ [ 2 ] \chi^{[2]} χ [ 2 ]
χ 2 = 1 τ ⟹ ξ = χ − τ / 2 , ξ 2 = 1 , χ = ξ τ / 2 , \chi^2=1_\tau\quad\Longrightarrow\quad\xi=\chi_{-\tau/2},\qquad\xi^2=1,\qquad\chi=\xi_{\tau/2}, χ 2 = 1 τ ⟹ ξ = χ − τ /2 , ξ 2 = 1 , χ = ξ τ /2 , contrary to the hypothesis. Here equality of primitive characters is understood on the idele class group, so it includes the local components at primes where squaring removes ramification. Positivity of 3 + 4 cos θ + cos 2 θ 3+4\cos\theta+\cos2\theta 3 + 4 cos θ + cos 2 θ
0 ≤ − 3 ζ K ′ ζ K ( σ ) − 4 Re L ′ L ( σ + i t , χ ) − Re L ′ L ( σ + 2 i t , χ [ 2 ] ) . 0 \le-3\frac{\zeta_K'}{\zeta_K}(\sigma)-4\operatorname{Re}\frac{L'}{L}(\sigma+it,\chi)-\operatorname{Re}\frac{L'}{L}(\sigma+2it,\chi^{[2]}). 0 ≤ − 3 ζ K ζ K ′ ( σ ) − 4 Re L L ′ ( σ + i t , χ ) − Re L L ′ ( σ + 2 i t , χ [ 2 ] ) . At a prime ramified for χ \chi χ 3 − 1 = 2 3-1=2 3 − 1 = 2 Lemma 3.2 , retaining the indicated zero and dropping the others,
4 σ − β ≤ 3 σ − 1 + 8 H . \frac{4}{\sigma-\beta}\le\frac{3}{\sigma-1}+8H. σ − β 4 ≤ σ − 1 3 + 8 H . Indeed, H ≥ 5 H\ge5 H ≥ 5
3 2 H ( 1 , 0 ) + 2 H + 1 2 H ( χ [ 2 ] , 2 t ) + 3 σ ≤ 9 2 H + 3 ≤ 11 2 H ≤ 8 H . \frac{3}{2}\mathcal{H}(1,0)+2H+\frac{1}{2}\mathcal{H}(\chi^{[2]},2t)+\frac{3}{\sigma}\le\frac{9}{2}H+3\le\frac{11}{2}H\le8H. 2 3 H ( 1 , 0 ) + 2 H + 2 1 H ( χ [ 2 ] , 2 t ) + σ 3 ≤ 2 9 H + 3 ≤ 2 11 H ≤ 8 H . Set σ = 1 + 1 / ( 16 H ) \sigma=1+1/(16H) σ = 1 + 1/ ( 16 H ) H H H 4 / ( 1 / 16 + ( 1 − β ) H ) ≤ 56 4/(1/16+(1-\beta)H)\le56 4/ ( 1/16 + ( 1 − β ) H ) ≤ 56 (19) . □
Proposition 3.4 (Real characters away from real zeros). Let ξ \xi ξ β + i u \beta+iu β + i u L ( s , ξ ) L(s,\xi) L ( s , ξ )
( 1 − β ) H ( ξ , u ) > 1 / 1000. (20) (1-\beta)\mathcal{H}(\xi,u)>1/1000. \tag*{(20)} ( 1 − β ) H ( ξ , u ) > 1/1000. (20) Proof. Suppose otherwise and set H = H ( ξ , u ) H=\mathcal{H}(\xi,u) H = H ( ξ , u ) κ = ( 1 − β ) H ≤ 1 / 1000 \kappa=(1-\beta)H\le1/1000 κ = ( 1 − β ) H ≤ 1/1000 b = ∣ u ∣ H > 0 b=|u|H>0 b = ∣ u ∣ H > 0 b ≤ 1 / 10 b\le1/10 b ≤ 1/10 σ = 1 + 1 / ( 4 H ) \sigma=1+1/(4H) σ = 1 + 1/ ( 4 H ) − ζ K ′ / ζ K − L ′ / L ( s , ξ ) -\zeta_K'/\zeta_K-L'/L(s,\xi) − ζ K ′ / ζ K − L ′ / L ( s , ξ ) ξ \xi ξ ξ = 1 \xi=1 ξ = 1 − ζ K ′ / ζ K -\zeta_K'/\zeta_K − ζ K ′ / ζ K Lemma 3.2 gives, with x = 1 / 4 + κ x=1/4+\kappa x = 1/4 + κ
2 x x 2 + b 2 ≤ 4 + 1 + H − 1 ≤ 6. \frac{2x}{x^2+b^2}\le4+1+H^{-1}\le6. x 2 + b 2 2 x ≤ 4 + 1 + H − 1 ≤ 6. But the left side is at least 2 ( 251 / 1000 ) / ( ( 251 / 1000 ) 2 + ( 1 / 10 ) 2 ) > 6 2(251/1000)/((251/1000)^2+(1/10)^2)>6 2 ( 251/1000 ) / (( 251/1000 ) 2 + ( 1/10 ) 2 ) > 6
If b ≥ 1 / 10 b\ge1/10 b ≥ 1/10 σ = 1 + 1 / ( 100 H ) \sigma=1+1/(100H) σ = 1 + 1/ ( 100 H ) (18) with ξ [ 2 ] = 1 \xi^{[2]}=1 ξ [ 2 ] = 1 H ( 1 , 2 u ) ≤ 2 H ( ξ , u ) \mathcal{H}(1,2u)\le2\mathcal{H}(\xi,u) H ( 1 , 2 u ) ≤ 2 H ( ξ , u ) 1 + 0 i 1+0i 1 + 0 i 2 u 2u 2 u H / 4 H/4 H /4 ξ \xi ξ u u u 4 H 4H 4 H 8 8 8 9 H / 2 9H/2 9 H /2
4 1 / 100 + κ ≤ 300 + 13 + 4 + 1 4 = 1269 4 . \frac{4}{1/100+\kappa}\le300+13+4+\frac{1}{4}=\frac{1269}{4}. 1/100 + κ 4 ≤ 300 + 13 + 4 + 4 1 = 4 1269 . The left side is at least 4000 / 11 > 1269 / 4 4000/11>1269/4 4000/11 > 1269/4
An embedding determinant estimate
In this section, we construct the embedding determinant used in § 5.1 . The evaluation points lie in a possibly nonnormal field E E E M M M M M M
Masser’s theorem and selected embeddings
In this subsection, we combine Masser’s zero estimate with the selected-embedding rank calculation. This gives the injectivity needed to choose a nonzero minor in § 4.3 .
Theorem 4.1 (Masser’s polynomial zero estimate). Let d , s ≥ 1 d,s \ge1 d , s ≥ 1 v 1 , … , v d ∈ C s v_1,\ldots,v_d \in\mathbb{C}^s v 1 , … , v d ∈ C s Γ \Gamma Γ d d d
μ ( Γ ) = min V ⊊ C s d − rank Z ( Γ ∩ V ) s − dim C V . \mu(\Gamma)=\min_{V\subsetneq\mathbb{C}^s}\frac{d-\operatorname{rank}_{\mathbb{Z}}(\Gamma\cap V)}{s-\dim_{\mathbb{C}}V}. μ ( Γ ) = V ⊊ C s min s − dim C V d − rank Z ( Γ ∩ V ) . The minimum ranges over proper complex linear subspaces V V V C s \mathbb{C}^s C s N ≥ 2 N\ge2 N ≥ 2 P P P ∑ j = 1 d u j v j \sum_{j=1}^{d}u_jv_j ∑ j = 1 d u j v j 0 ≤ u j ≤ N − 1 0\le u_j\le N-1 0 ≤ u j ≤ N − 1
deg P ≥ ( N − 1 s ) μ ( Γ ) . (21) \deg P\ge\left(\frac{N-1}{s}\right)^{\mu(\Gamma)}. \tag*{(21)} deg P ≥ ( s N − 1 ) μ ( Γ ) . (21) This is the polynomial form of [15 Theorem 1] , as used in [9 Lemma 6] and [1 Theorem 4.1] . It is the only polynomial zero estimate used below. Its application is over C \mathbb{C} C
Lemma 4.2 (Selected-embedding exponent). Let E / Q E/\mathbb{Q} E / Q d d d S S S 1 ≤ s ≤ d 1\le s\le d 1 ≤ s ≤ d E ↪ C E\hookrightarrow\mathbb{C} E ↪ C ω 1 , … , ω d ∈ E \omega_1,\ldots,\omega_d\in E ω 1 , … , ω d ∈ E ( σ ω j ) σ ∈ S (\sigma\omega_j)_{\sigma\in S} ( σ ω j ) σ ∈ S μ ( Γ ) = d / s \mu(\Gamma)=d/s μ ( Γ ) = d / s N ≥ 2 N\ge2 N ≥ 2 D ≥ 0 D\ge0 D ≥ 0 D D D
( σ ∑ j = 1 d u j ω j ) σ ∈ S , u j ∈ { 0 , … , N − 1 } , \left(\sigma\sum_{j=1}^{d}u_j\omega_j\right)_{\sigma\in S},\qquad u_j\in\{0,\ldots,N-1\}, ( σ j = 1 ∑ d u j ω j ) σ ∈ S , u j ∈ { 0 , … , N − 1 } , if D < ( ( N − 1 ) / s ) d / s D<((N-1)/s)^{d/s} D < (( N − 1 ) / s ) d / s
This is the selected-embedding calculation in [1 Proposition 4.2 and its proof] . It applies to any set of embeddings. We give the proof for E E E
Proof. The embedding map is injective on E E E d d d U ⊂ E U\subset E U ⊂ E r r r u 1 , … , u r u_1,\ldots,u_r u 1 , … , u r d d d r r r r r r B B B
Let G G G E E E d d d
rank ( σ u j ) σ ∈ S , j ≥ 1 ∣ G ∣ ∑ g ∈ G ∣ g S ∩ B ∣ = s r d . \operatorname{rank}(\sigma u_j)_{\sigma\in S,j}\ge\frac{1}{|G|}\sum_{g\in G}|gS\cap B|=\frac{sr}{d}. rank ( σ u j ) σ ∈ S , j ≥ ∣ G ∣ 1 g ∈ G ∑ ∣ g S ∩ B ∣ = d sr . For a complex subspace V V V v v v U U U Γ ∩ V \Gamma\cap V Γ ∩ V r = rank Z ( Γ ∩ V ) r=\operatorname{rank}_{\mathbb{Z}}(\Gamma\cap V) r = rank Z ( Γ ∩ V ) V V V r ≤ d v / s r\le dv/s r ≤ d v / s μ ( Γ ) ≥ d / s \mu(\Gamma)\ge d/s μ ( Γ ) ≥ d / s V = 0 V=0 V = 0 Theorem 4.1 . □
Weighted monomial sums
In this subsection, we compute finite weighted monomial sums for the determinant of § 4.3 . The resulting ratios give the explicit prime-sum constants in § 5.1 .
For α = ( α 1 , … , α s ) ∈ Z ≥ 0 s \alpha=(\alpha_1,\ldots,\alpha_s)\in\mathbb{Z}_{\geq0}^s α = ( α 1 , … , α s ) ∈ Z ≥ 0 s ∣ α ∣ = ∑ i = 1 s α i |\alpha|=\sum_{i=1}^s\alpha_i ∣ α ∣ = ∑ i = 1 s α i z α = ∏ i = 1 s z i α i z^\alpha=\prod_{i=1}^s z_i^{\alpha_i} z α = ∏ i = 1 s z i α i H 0 ≥ 2 H_0\geq2 H 0 ≥ 2 D ≥ 2 H 0 D\geq2H_0 D ≥ 2 H 0 s ≥ 1 s\geq1 s ≥ 1
A = { α ∈ Z ≥ 0 s : α 1 + H 0 ( α 2 + ⋯ + α s ) ≤ D } , A = ∣ A ∣ , \mathcal{A}=\{\alpha\in\mathbb{Z}_{\geq0}^s:\alpha_1+H_0(\alpha_2+\cdots+\alpha_s)\leq D\},\qquad A=|\mathcal{A}|, A = { α ∈ Z ≥ 0 s : α 1 + H 0 ( α 2 + ⋯ + α s ) ≤ D } , A = ∣ A ∣ , S 1 = ∑ α ∈ A α 1 , W = ∑ α ∈ A ∣ α ∣ , E p = ∑ α ∈ A ⌊ α 1 p ⌋ . S_1=\sum_{\alpha\in\mathcal{A}}\alpha_1,\qquad W=\sum_{\alpha\in\mathcal{A}}|\alpha|,\qquad E_p=\sum_{\alpha\in\mathcal{A}}\left\lfloor\frac{\alpha_1}{p}\right\rfloor. S 1 = α ∈ A ∑ α 1 , W = α ∈ A ∑ ∣ α ∣ , E p = α ∈ A ∑ ⌊ p α 1 ⌋ . The following finite sums replace the fixed-dimension asymptotic monomial estimates in [1 Lemma 6.1] ; we use them in the prime-sum estimate of Lemma 5.2 .
Lemma 4.3 (Monomial ratios). The following bounds hold without asymptotic error:
W S 1 ≤ 1 + s − 1 H 0 , A S 1 ≤ s + 1 D − H 0 ≤ 2 ( s + 1 ) D , E p ≥ S 1 p − A . (22) \frac{W}{S_1}\leq1+\frac{s-1}{H_0},\qquad\frac{A}{S_1}\leq\frac{s+1}{D-H_0}\leq\frac{2(s+1)}{D},\qquad E_p\geq\frac{S_1}{p}-A. \tag*{(22)} S 1 W ≤ 1 + H 0 s − 1 , S 1 A ≤ D − H 0 s + 1 ≤ D 2 ( s + 1 ) , E p ≥ p S 1 − A . (22) Also A ≤ ( D + 1 ) s A\leq(D+1)^s A ≤ ( D + 1 ) s
Proof. Write α 1 = H 0 b 1 + r \alpha_1=H_0b_1+r α 1 = H 0 b 1 + r 0 ≤ r < H 0 0\leq r<H_0 0 ≤ r < H 0 q r = ⌊ ( D − r ) / H 0 ⌋ q_r=\lfloor(D-r)/H_0\rfloor q r = ⌊( D − r ) / H 0 ⌋ r r r s s s b 1 , α 2 , … , α s b_1,\alpha_2,\ldots,\alpha_s b 1 , α 2 , … , α s q r q_r q r q r / ( s + 1 ) q_r/(s+1) q r / ( s + 1 ) q r − b 1 − α 2 − ⋯ − α s q_r-b_1-\alpha_2-\cdots-\alpha_s q r − b 1 − α 2 − ⋯ − α s
Summing over residues, write T T T b 1 b_1 b 1 R ≥ 0 R\geq0 R ≥ 0 S 1 = H 0 T + R S_1=H_0T+R S 1 = H 0 T + R W = S 1 + ( s − 1 ) T W=S_1+(s-1)T W = S 1 + ( s − 1 ) T
In each residue class the mean of α 1 \alpha_1 α 1 H 0 q r / ( s + 1 ) + r ≥ ( D − H 0 ) / ( s + 1 ) H_0q_r/(s+1)+r\geq(D-H_0)/(s+1) H 0 q r / ( s + 1 ) + r ≥ ( D − H 0 ) / ( s + 1 ) ⌊ x ⌋ ≥ x − 1 \lfloor x\rfloor\geq x-1 ⌊ x ⌋ ≥ x − 1
Frobenius congruences and the determinant
In this subsection, we compare the archimedean size and Frobenius divisibility of a nonzero evaluation determinant. We use the injectivity of § 4.1 and the monomial sums of § 4.2 ; § 5.1 then specializes the resulting inequality.
Let E / Q E/\mathbb{Q} E / Q d ≥ 2 d\geq2 d ≥ 2 M M M m = [ M : Q ] m=[M:\mathbb{Q}] m = [ M : Q ] S S S s < d s<d s < d E E E M M M p p p M M M C p ⊂ Gal ( M / Q ) C_p\subset\operatorname{Gal}(M/\mathbb{Q}) C p ⊂ Gal ( M / Q )
δ p = ∣ { g ∈ C p : g ∣ E ∈ S } ∣ ∣ C p ∣ . (23) \delta_p=\frac{|\{g\in C_p:g|_E\in S\}|}{|C_p|}. \tag*{(23)} δ p = ∣ C p ∣ ∣ { g ∈ C p : g ∣ E ∈ S } ∣ . (23) Set δ p = 0 \delta_p=0 δ p = 0 λ E \lambda_E λ E d d d E E E E E E Q \mathbb{Q} Q
One can take λ E ≤ D E \lambda_E\leq\sqrt{D_E} λ E ≤ D E ( r 1 , r 2 ) (r_1,r_2) ( r 1 , r 2 ) E E E r 1 r_1 r 1 r 2 r_2 r 2 λ 1 ≤ ⋯ ≤ λ d \lambda_1\leq\cdots\leq\lambda_d λ 1 ≤ ⋯ ≤ λ d O E \mathcal{O}_E O E λ j \lambda_j λ j j j j 2 r 1 π r 2 2^{r_1}\pi^{r_2} 2 r 1 π r 2 O E \mathcal{O}_E O E 2 − r 2 D E 2^{-r_2}\sqrt{D_E} 2 − r 2 D E
∏ j = 1 d λ j ≤ ( 2 / π ) r 2 D E ≤ D E . \prod_{j=1}^{d}\lambda_j\leq(2/\pi)^{r_2}\sqrt{D_E}\leq\sqrt{D_E}. j = 1 ∏ d λ j ≤ ( 2/ π ) r 2 D E ≤ D E . Every nonzero algebraic integer has house at least one, by its nonzero integer norm. Thus the last minimum, and hence a house bound for d d d D E \sqrt{D_E} D E
Lemma 4.4 (Nonnormal-field determinant comparison). Let H 0 ≥ 2 H_0 \ge2 H 0 ≥ 2 D ≥ 2 H 0 D \ge2H_0 D ≥ 2 H 0 S = ( σ 1 , … , σ s ) S = (\sigma_1,\ldots,\sigma_s) S = ( σ 1 , … , σ s ) σ 1 \sigma_1 σ 1 z σ z_\sigma z σ σ ∈ S \sigma\in S σ ∈ S z i = z σ i z_i = z_{\sigma_i} z i = z σ i 1 ≤ i ≤ s 1 \le i \le s 1 ≤ i ≤ s
N = 1 + ⌈ 2 s ( D + 1 ) s / d ⌉ , R = d λ E N . N = 1 + \left\lceil2s(D+1)^{s/d} \right\rceil,\qquad\mathcal{R} = d\lambda_E N. N = 1 + ⌈ 2 s ( D + 1 ) s / d ⌉ , R = d λ E N . Then
∑ H 0 ≤ p ≤ D p unramified in M δ p E p log p ≤ W log R + A 2 log A . (24) \sum_{\substack{H_0 \le p \le D\\p\ \text{unramified in }M}} \delta_p E_p \log p \le W\log\mathcal{R} + \frac{A}{2}\log A. \tag*{(24)} H 0 ≤ p ≤ D p unramified in M ∑ δ p E p log p ≤ W log R + 2 A log A . (24) The same upper bound holds if the prime sum is restricted to any smaller cutoff X ≤ D X \le D X ≤ D
We compare the archimedean size and Frobenius divisibility of a nonzero determinant, following OpenAI [20 ] and the explicit exposition of Banks [2 ] ; related determinant comparisons appear in [8 , 3 , 10 ] . The nonzero minor here is supplied by Masser’s estimate and Lemma 4.2. The triangular polynomial substitutions are those of [1 Lemma 5.2] . The quantitative adaptation here allows the points to lie in E E E M M M
Proof. Use the points θ u = ∑ u j ω j \theta_u = \sum u_j\omega_j θ u = ∑ u j ω j 0 ≤ u j < N 0 \le u_j < N 0 ≤ u j < N ω j ∈ O E \omega_j \in\mathcal{O}_E ω j ∈ O E λ E \lambda_E λ E R \mathcal{R} R
By Lemma 4.2, evaluation on the whole box is injective on the polynomials spanned by z α z^\alpha z α α ∈ A \alpha\in\mathcal{A} α ∈ A D D D ( ( N − 1 ) / s ) d / s ≥ 2 d / s ( D + 1 ) > D ((N-1)/s)^{d/s} \ge2^{d/s}(D+1) > D (( N − 1 ) / s ) d / s ≥ 2 d / s ( D + 1 ) > D A \mathcal{A} A A A A D ∈ O M \mathcal{D} \in\mathcal{O}_M D ∈ O M
Hadamard’s inequality at every embedding of M M M
1 m log ∣ N M / Q D ∣ ≤ W log R + A 2 log A . (25) \frac{1}{m}\log\left|N_{M/\mathbb{Q}}\mathcal{D}\right| \le W\log\mathcal{R} + \frac{A}{2}\log A. \tag*{(25)} m 1 log N M / Q D ≤ W log R + 2 A log A . (25) Indeed the row indexed by α \alpha α A R ∣ α ∣ \sqrt{A}\mathcal{R}^{|\alpha|} A R ∣ α ∣
For a fixed embedding σ ∈ S \sigma\in S σ ∈ S Q p , σ ⊂ O M Q_{p,\sigma} \subset\mathcal{O}_M Q p , σ ⊂ O M P P P p p p Frob P ∣ E = σ \operatorname{Frob}_{P|E} = \sigma Frob P ∣ E = σ O M \mathcal{O}_M O M θ ∈ O E \theta\in\mathcal{O}_E θ ∈ O E
θ p ≡ σ ( θ ) ( m o d Q p , σ ) , ∑ σ ∈ S log N Q p , σ = m δ p log p . \theta^p \equiv\sigma(\theta)\pmod{Q_{p,\sigma}},\qquad\sum_{\sigma\in S}\log NQ_{p,\sigma}=m\delta_p\log p. θ p ≡ σ ( θ ) ( mod Q p , σ ) , σ ∈ S ∑ log N Q p , σ = m δ p log p . Let f p = [ O M / P : F p ] f_p = [\mathcal{O}_M/P:\mathbb{F}_p] f p = [ O M / P : F p ] P P P p p p F p = Z / p Z \mathbb{F}_p = \mathbb{Z}/p\mathbb{Z} F p = Z / p Z p p p C p C_p C p m / ( f p ∣ C p ∣ ) m/(f_p|C_p|) m / ( f p ∣ C p ∣ ) p f p p^{f_p} p f p Q p , σ Q_{p,\sigma} Q p , σ
For p ≥ H 0 p \ge H_0 p ≥ H 0 σ \sigma σ α 1 = p b + v \alpha_1 = pb+v α 1 = p b + v
z 1 v ( z 1 p − z σ ) b z 2 α 2 ⋯ z s α s . z_1^v(z_1^p-z_\sigma)^b z_2^{\alpha_2}\cdots z_s^{\alpha_s}. z 1 v ( z 1 p − z σ ) b z 2 α 2 ⋯ z s α s . For σ ≠ σ 1 \sigma\ne\sigma_1 σ = σ 1 j j j α 1 − p j \alpha_1-pj α 1 − p j α 1 + H 0 ∑ i > 1 α i − ( p − H 0 ) j \alpha_1+H_0\sum_{i>1}\alpha_i-(p-H_0)j α 1 + H 0 ∑ i > 1 α i − ( p − H 0 ) j σ = σ 1 \sigma=\sigma_1 σ = σ 1 α 1 − ( p − 1 ) j \alpha_1-(p-1)j α 1 − ( p − 1 ) j A \mathcal{A} A p = H 0 p=H_0 p = H 0 ( p , σ ) (p,\sigma) ( p , σ ) D \mathcal{D} D Q p , σ E \mathbb{Q}_{p,\sigma}^{E} Q p , σ E D ∈ Q p , σ E \mathcal{D}\in\mathbb{Q}_{p,\sigma}^{E} D ∈ Q p , σ E
1 m log ∣ N M / Q D ∣ ≥ ∑ H 0 ≤ p ≤ D p u n r a m i f i e d δ p E p log p . (26) \frac{1}{m}\log\left|\mathrm{N}_{M/\mathbb{Q}}\mathcal{D}\right|\geq\sum_{\substack{H_0\leq p\leq D\\p\ \mathrm{unramified}}}\delta_p E_p\log p. \tag*{(26)} m 1 log N M / Q D ≥ H 0 ≤ p ≤ D p unramified ∑ δ p E p log p . (26) Combine (25) and (26) . The normal-closure factor m m m M M M
Real zeros over arbitrary number fields
In this section, we apply the determinant estimate of § 4 to an auxiliary quadratic extension and compare the resulting prime sum with the bias from a real zeta zero. The cutoff in § 5.4 proves Theorem 1.1 , which § 6 transfers to Hecke characters.
Small independent integers and an explicit prime sum
In this subsection, we combine small independent algebraic integers with the determinant estimate of § 4.3 to obtain an explicit prime-sum bound. The polynomial degree and prime cutoff are chosen separately; § 5.3 uses the resulting bound.
Lemma 5.1 (Small independent integers). For a degree-d d d E E E d d d D E 2 / d D_E^{2/d} D E 2/ d
log λ E ≤ 2 d log D E . (27) \log\lambda_E\leq\frac{2}{d}\log D_E. \tag*{(27)} log λ E ≤ d 2 log D E . (27) Proof. This is the explicit small-integer bound of [6 Proposition 1] ; we recall its argument to keep constants visible. Let λ 1 ≤ ⋯ ≤ λ d \lambda_1\leq\cdots\leq\lambda_d λ 1 ≤ ⋯ ≤ λ d D E \sqrt{D_E} D E h = ⌊ d / 2 ⌋ + 1 h=\lfloor d/2\rfloor+1 h = ⌊ d /2 ⌋ + 1 λ h d − h + 1 ∏ j = 1 h − 1 λ j ≤ D E \lambda_h^{d-h+1}\prod_{j=1}^{h-1}\lambda_j\leq\sqrt{D_E} λ h d − h + 1 ∏ j = 1 h − 1 λ j ≤ D E d − h + 1 ≥ d / 2 d-h+1\geq d/2 d − h + 1 ≥ d /2 λ h ≤ D E 1 / d \lambda_h\leq D_E^{1/d} λ h ≤ D E 1/ d
Let U ⊂ E U\subset E U ⊂ E h h h U U U E E E f : E → Q f:E\to\mathbb{Q} f : E → Q U U = { u v : u , v ∈ U } UU=\{uv:u,v\in U\} U U = { uv : u , v ∈ U } E ∗ = Hom Q ( E , Q ) E^*=\operatorname{Hom}_{\mathbb{Q}}(E,\mathbb{Q}) E ∗ = Hom Q ( E , Q )
U ⟶ E ∗ , u ⟼ [ v ⟼ f ( u v ) ] U\longrightarrow E^*,\qquad u\longmapsto[v\longmapsto f(uv)] U ⟶ E ∗ , u ⟼ [ v ⟼ f ( uv )] is injective, because multiplication by a nonzero field element is invertible. Its image lies in Ann E ∗ ( U ) = { ℓ ∈ E ∗ : ℓ ∣ U = 0 } \operatorname{Ann}_{E^*}(U)=\{\ell\in E^*:\ell|_U=0\} Ann E ∗ ( U ) = { ℓ ∈ E ∗ : ℓ ∣ U = 0 } d − h < h d-h<h d − h < h
Select d d d D E 2 / d D_E^{2/d} D E 2/ d
Lemma 5.2 (Refined embedding prime sum). Let E / Q E/\mathbb{Q} E / Q d ≥ 2 d\geq2 d ≥ 2 S S S (23) . Put B E = log D E B_E=\log D_E B E = log D E
one has
∑ p ≤ X δ p log p p ≤ ( 1 − 15 16 d ) log X + 4 d B E + 32 log ( 2 d ) . (28) \sum_{p\leq X}\delta_p\frac{\log p}{p}\leq\left(1-\frac{15}{16d}\right)\log X+\frac{4}{d}B_E+32\log(2d). \tag*{(28)} p ≤ X ∑ δ p p log p ≤ ( 1 − 16 d 15 ) log X + d 4 B E + 32 log ( 2 d ) . (28) Proof. In the determinant construction choose
s = d − 1 , H 0 = 16 d 2 , D = 16 d 2 X , N = 1 + ⌈ 2 s ( D + 1 ) s / d ⌉ . s=d-1,\qquad H_0=16d^2,\qquad D=16d^2X,\qquad N=1+\left\lceil2s(D+1)^{s/d}\right\rceil. s = d − 1 , H 0 = 16 d 2 , D = 16 d 2 X , N = 1 + ⌈ 2 s ( D + 1 ) s / d ⌉ . Use a spanning set from Lemma 5.1 , and apply Lemma 4.4 , restricted to primes H 0 ≤ p ≤ X H_0\le p\le X H 0 ≤ p ≤ X D D D ν = 1 + ( s − 1 ) / H 0 \nu=1+(s-1)/H_0 ν = 1 + ( s − 1 ) / H 0 ν s / d ≤ 1 − 15 / ( 16 d ) \nu s/d\le1-15/(16d) ν s / d ≤ 1 − 15/ ( 16 d )
2 d D ϑ ( X ) ≤ 1 2 d . \frac{2d}{D}\vartheta(X)\le\frac{1}{2d}. D 2 d ϑ ( X ) ≤ 2 d 1 . The determinant error is at most
A 2 S 1 log A ≤ d s D log ( D + 1 ) ≤ 1 4 . \frac{A}{2S_1}\log A\le\frac{ds}{D}\log(D+1)\le\frac{1}{4}. 2 S 1 A log A ≤ D d s log ( D + 1 ) ≤ 4 1 . Here D ≥ 16 d 4 D\ge16d^4 D ≥ 16 d 4 X ≥ d 2 X\ge d^2 X ≥ d 2 log ( D + 1 ) ≤ D \log(D+1)\le\sqrt{D} log ( D + 1 ) ≤ D 1 / ( 2 d ) + 1 / 4 ≤ 1 / 2 1/(2d)+1/4\le1/2 1/ ( 2 d ) + 1/4 ≤ 1/2
Writing T = ( D + 1 ) s / d ≥ 1 T=(D+1)^{s/d}\ge1 T = ( D + 1 ) s / d ≥ 1
N ≤ 2 s T + 2 ≤ 2 d T , log R ≤ s d log ( D + 1 ) + log ( 2 d 2 ) + 2 d B E . N\le2sT+2\le2dT,\qquad\log R\le\frac{s}{d}\log(D+1)+\log(2d^2)+\frac{2}{d}B_E. N ≤ 2 s T + 2 ≤ 2 d T , log R ≤ d s log ( D + 1 ) + log ( 2 d 2 ) + d 2 B E . Thus the point-height contribution is at most
ν { s d log ( D + 1 ) + log ( 2 d 2 ) + 2 d B E } . \nu\left\{\frac{s}{d}\log(D+1)+\log(2d^2)+\frac{2}{d}B_E\right\}. ν { d s log ( D + 1 ) + log ( 2 d 2 ) + d 2 B E } . Use D + 1 ≤ 32 d 2 X D+1\le32d^2X D + 1 ≤ 32 d 2 X ν ≤ 2 \nu\le2 ν ≤ 2 ν s / d ≤ 1 \nu s/d\le1 ν s / d ≤ 1 H 0 H_0 H 0 4 ( 1 + log H 0 ) 4(1+\log H_0) 4 ( 1 + log H 0 ) 4 B E / d 4B_E/d 4 B E / d 5 + 23 log 2 + 14 log d ≤ 24 + 14 log d ≤ 32 log ( 2 d ) 5+23\log2+14\log d\le24+14\log d\le32\log(2d) 5 + 23 log 2 + 14 log d ≤ 24 + 14 log d ≤ 32 log ( 2 d ) 1 / 2 < log 2 < 7 / 10 1/2<\log2<7/10 1/2 < log 2 < 7/10 d ≥ 2 d\ge2 d ≥ 2
An auxiliary quadratic field and degree-one primes
In this subsection, we choose the auxiliary quadratic field and relate the omitted embedding to degree-one primes of K K K § 5.1 into the lower comparison used in § 5.3 .
Let K / Q K/\mathbb{Q} K / Q n n n a a a a ∉ K \sqrt{a}\notin K a ∈ / K ⌊ log 2 n ⌋ + 1 \lfloor\log_2 n\rfloor+1 ⌊ log 2 n ⌋ + 1 K K K n n n a ≤ 2 n a\le2n a ≤ 2 n
Set
E = K ( a ) , d = 2 n , B E = log D E , E=K(\sqrt{a}),\qquad d=2n,\qquad B_E=\log D_E, E = K ( a ) , d = 2 n , B E = log D E , 2 log D K ≤ B E ≤ 4 log D K + n log 8. (29) 2\log D_K\le B_E\le4\log D_K+n\log8. \tag*{(29)} 2 log D K ≤ B E ≤ 4 log D K + n log 8. (29) The lower bound is the discriminant tower formula for E / K E/K E / K K K K Q ( a ) \mathbb{Q}(\sqrt{a}) Q ( a ) E E E D K 2 D Q ( a ) n D_K^2D_{\mathbb{Q}(\sqrt{a})}^n D K 2 D Q ( a ) n a = 2 a=2 a = 2 b b b a a a Q ( b ) ⊂ K \mathbb{Q}(\sqrt{b})\subset K Q ( b ) ⊂ K log D K ≥ ( n / 2 ) log b \log D_K\ge(n/2)\log b log D K ≥ ( n /2 ) log b a < 2 b a<2b a < 2 b log D Q ( a ) ≤ log ( 4 a ) ≤ log 8 + ( 2 / n ) log D K \log D_{\mathbb{Q}(\sqrt{a})}\le\log(4a)\le\log8+(2/n)\log D_K log D Q ( a ) ≤ log ( 4 a ) ≤ log 8 + ( 2/ n ) log D K ι ≠ 1 \iota\ne1 ι = 1 K K K E E E S S S d d d E E E ι \iota ι s = d − 1 s=d-1 s = d − 1
Lemma 5.3 (Fixed-point comparison). For p p p E E E r K ( p ) r_K(p) r K ( p ) K K K p p p
δ p = 1 − 1 η a ( p ) = − 1 r K ( p ) n , η a = χ Q ( a ) / Q . (30) \delta_p = 1 - \mathbf{1}_{\eta_a(p)=-1}\frac{r_K(p)}{n}, \qquad\eta_a = \chi_{\mathbb{Q}(\sqrt{a})/\mathbb{Q}}. \tag*{(30)} δ p = 1 − 1 η a ( p ) = − 1 n r K ( p ) , η a = χ Q ( a ) / Q . (30) In particular,
1 − δ p ≤ r K ( p ) n . (31) 1-\delta_p \le\frac{r_K(p)}{n}. \tag*{(31)} 1 − δ p ≤ n r K ( p ) . (31) Consequently, for X ≥ 2 X \ge2 X ≥ 2
∑ p ≤ X δ p log p p ≥ log X − 6 − 1 n ∑ N p ≤ X log N p N p − ∑ p ∣ D E log p p . (32) \sum_{p\le X}\delta_p\frac{\log p}{p} \ge\log X-6-\frac{1}{n}\sum_{N\mathfrak{p}\le X}\frac{\log N\mathfrak{p}}{N\mathfrak{p}}-\sum_{p\mid D_E}\frac{\log p}{p}. \tag*{(32)} p ≤ X ∑ δ p p log p ≥ log X − 6 − n 1 N p ≤ X ∑ N p log N p − p ∣ D E ∑ p log p . (32) Proof. If g ∣ E = ι g|_E=\iota g ∣ E = ι g ∣ K = 1 g|_K=1 g ∣ K = 1 K K K r K ( p ) / n r_K(p)/n r K ( p ) / n K K K
The restriction of every member of the class to Q ( a ) \mathbb{Q}(\sqrt{a}) Q ( a ) η a ( p ) \eta_a(p) η a ( p ) − 1 -1 − 1 K K K g ∣ E = ι g|_E=\iota g ∣ E = ι (30) and (31) .
Primes unramified in E E E D E D_E D E (31) , use (14) , and bound the degree-one prime sum by the sum over all prime ideals of K K K
Ramification and the final prime-sum comparison
In this subsection, we bound the ramification contribution and retain the exact real-zero term in the logarithmic derivative. Combining Sections 5.1 and 5.2 gives the comparison and error bound used to choose the cutoff in Section 5.4.
Lemma 5.4 (Logarithmic ramification bound). For a positive integer discriminant D E > 1 D_E>1 D E > 1 B E = log D E B_E=\log D_E B E = log D E
∑ p ∣ D E log p p ≤ 5 + 4 log ( 2 + B E ) . \sum_{p\mid D_E}\frac{\log p}{p}\le5+4\log(2+B_E). p ∣ D E ∑ p log p ≤ 5 + 4 log ( 2 + B E ) . Proof. Split at Y = max ( 2 , B E ) Y=\max(2,B_E) Y = max ( 2 , B E ) ϑ ( u ) ≤ 4 u \vartheta(u)\le4u ϑ ( u ) ≤ 4 u Y Y Y 4 + 4 log Y 4+4\log Y 4 + 4 log Y Y Y Y Y − 1 ∑ p ∣ D E log p ≤ B E / Y ≤ 1 Y^{-1}\sum_{p\mid D_E}\log p\le B_E/Y\le1 Y − 1 ∑ p ∣ D E log p ≤ B E / Y ≤ 1
Throughout Sections 5.3–5.4, put
Δ = 1 − β , 0 < β < 1 , ζ K ( β ) = 0 , L = log D K , H = H K + ( 1 , 0 ) , A K = 1 + 1 2 H K ( 1 , 0 ) . \Delta=1-\beta,\qquad0<\beta<1,\qquad\zeta_K(\beta)=0,\qquad L=\log D_K,\qquad H=H_K^+(1,0),\qquad A_K=1+\frac{1}{2}H_K(1,0). Δ = 1 − β , 0 < β < 1 , ζ K ( β ) = 0 , L = log D K , H = H K + ( 1 , 0 ) , A K = 1 + 2 1 H K ( 1 , 0 ) . Choose E = K ( a ) E=K(\sqrt{a}) E = K ( a ) (29) ; put d = 2 n d=2n d = 2 n B E = log D E B_E=\log D_E B E = log D E α 0 = 15 / 16 \alpha_0=15/16 α 0 = 15/16
Lemma 5.5 (Unrounded real-zero term). For an integer X ≥ d 2 X\ge d^2 X ≥ d 2 ℓ = log X \ell=\log X ℓ = log X 0 < u ≤ ℓ 0<u\le\ell 0 < u ≤ ℓ
∑ N p ≤ X log N p N p ≤ e u { Δ ℓ 2 u ( u + Δ ℓ ) + A K } . (33) \sum_{N\mathfrak{p}\le X}\frac{\log N\mathfrak{p}}{N\mathfrak{p}}\le e^u\left\{\frac{\Delta\ell^2}{u(u+\Delta\ell)}+A_K\right\}. \tag*{(33)} N p ≤ X ∑ N p log N p ≤ e u { u ( u + Δ ℓ ) Δ ℓ 2 + A K } . (33) Consequently,
α 0 ℓ ≤ 2 e u Δ ℓ 2 u ( u + Δ ℓ ) + E E ( u ) , (34) \alpha_0\ell\le\frac{2e^u\Delta\ell^2}{u(u+\Delta\ell)}+\mathcal{E}_E(u), \tag*{(34)} α 0 ℓ ≤ u ( u + Δ ℓ ) 2 e u Δ ℓ 2 + E E ( u ) , (34) where
E E ( u ) = 4 B E + 32 d log ( 2 d ) + 11 d + 4 d log ( 2 + B E ) + 2 e u A K . (35) \mathcal{E}_E(u)=4B_E+32d\log(2d)+11d+4d\log(2+B_E)+2e^uA_K. \tag*{(35)} E E ( u ) = 4 B E + 32 d log ( 2 d ) + 11 d + 4 d log ( 2 + B E ) + 2 e u A K . (35) Proof. Take h = u / ℓ ∈ ( 0 , 1 ] h=u/\ell\in(0,1] h = u / ℓ ∈ ( 0 , 1 ] Lemma 3.2 , retaining the given zero, gives
− ζ K ′ ζ K ( 1 + h ) ≤ 1 h − 1 h + Δ + 1 + 1 2 H K ( 1 , 0 ) = Δ h ( h + Δ ) + A K . -\frac{\zeta'_K}{\zeta_K}(1+h) \le\frac{1}{h}-\frac{1}{h+\Delta}+1+\frac{1}{2}\mathcal{H}_K(1,0)=\frac{\Delta}{h(h+\Delta)}+A_K. − ζ K ζ K ′ ( 1 + h ) ≤ h 1 − h + Δ 1 + 1 + 2 1 H K ( 1 , 0 ) = h ( h + Δ ) Δ + A K . Here 1 < 1 + h ≤ 2 1<1+h\le2 1 < 1 + h ≤ 2 1 / ( 1 + h ) ≤ 1 1/(1+h)\le1 1/ ( 1 + h ) ≤ 1 ( N p ) − h ≥ e − u (N\mathfrak{p})^{-h}\ge e^{-u} ( N p ) − h ≥ e − u N p ≤ X N\mathfrak{p}\le X N p ≤ X (33) . Now combine (32) , Lemma 5.4 , and Lemma 5.2 . Since d / n = 2 d/n=2 d / n = 2
α 0 ℓ ≤ d n ∑ N p ≤ X log N p N p + 4 B E + 32 d log ( 2 d ) + 6 d + d { 5 + 4 log ( 2 + B E ) } . \alpha_0\ell\le\frac{d}{n}\sum_{N\mathfrak{p}\le X}\frac{\log N\mathfrak{p}}{N\mathfrak{p}}+4B_E+32d\log(2d)+6d+d\{5+4\log(2+B_E)\}. α 0 ℓ ≤ n d N p ≤ X ∑ N p log N p + 4 B E + 32 d log ( 2 d ) + 6 d + d { 5 + 4 log ( 2 + B E )} . Insert (33) , retaining the denominator h ( h + Δ ) h(h+\Delta) h ( h + Δ )
Lemma 5.6 (A uniform error bound at u = 2 u=2 u = 2 For the preceding fields,
E E ( 2 ) ≤ 80 H . (36) \mathcal{E}_E(2)\le80H. \tag*{(36)} E E ( 2 ) ≤ 80 H . (36) Proof. The series for e e e
e ≤ 65 24 + 1 100 < 11 4 , e 2 < 8. e\le\frac{65}{24}+\frac{1}{100}<\frac{11}{4},\qquad e^2<8. e ≤ 24 65 + 100 1 < 4 11 , e 2 < 8. Indeed the tail starting at 1 / 5 ! 1/5! 1/5 ! ( 1 / 120 ) ∑ j ≥ 0 6 − j = 1 / 100 (1/120)\sum_{j\ge0}6^{-j}=1/100 ( 1/120 ) ∑ j ≥ 0 6 − j = 1/100 log ( 2 + B E ) ≤ log ( 2 d ) + B E / d \log(2+B_E)\le\log(2d)+B_E/d log ( 2 + B E ) ≤ log ( 2 d ) + B E / d
E E ( 2 ) ≤ 8 B E + 36 d log ( 2 d ) + 11 d + 8 H K ( 1 , 0 ) + 16. \mathcal{E}_E(2)\le8B_E+36d\log(2d)+11d+8\mathcal{H}_K(1,0)+16. E E ( 2 ) ≤ 8 B E + 36 d log ( 2 d ) + 11 d + 8 H K ( 1 , 0 ) + 16. Write U = n log ( 2 n ) U=n\log(2n) U = n log ( 2 n ) B E ≤ 4 L + 3 n log 2 B_E\le4L+3n\log2 B E ≤ 4 L + 3 n log 2 d = 2 n d=2n d = 2 n
E E ( 2 ) ≤ 32 L + 72 U + ( 96 log 2 + 22 ) n + 8 H K ( 1 , 0 ) + 16. \mathcal{E}_E(2)\le32L+72U+(96\log2+22)n+8\mathcal{H}_K(1,0)+16. E E ( 2 ) ≤ 32 L + 72 U + ( 96 log 2 + 22 ) n + 8 H K ( 1 , 0 ) + 16. Now H = H K ( 1 , 0 ) + U H=\mathcal{H}_K(1,0)+U H = H K ( 1 , 0 ) + U H K ( 1 , 0 ) ≥ L + 4 n + 1 \mathcal{H}_K(1,0)\ge L+4n+1 H K ( 1 , 0 ) ≥ L + 4 n + 1
80 H − E E ( 2 ) ≥ 72 H K ( 1 , 0 ) − 32 L + 8 U − ( 96 log 2 + 22 ) n − 16 80H-\mathcal{E}_E(2)\ge72\mathcal{H}_K(1,0)-32L+8U-(96\log2+22)n-16 80 H − E E ( 2 ) ≥ 72 H K ( 1 , 0 ) − 32 L + 8 U − ( 96 log 2 + 22 ) n − 16 ≥ 40 L + 8 U + ( 266 − 96 log 2 ) n + 56 > 0. \ge40L+8U+(266-96\log2)n+56>0. ≥ 40 L + 8 U + ( 266 − 96 log 2 ) n + 56 > 0. Choice of the final prime cutoff
In this subsection, we choose the logarithmic-derivative parameter and an integer prime cutoff in the comparison of § 5.3 , proving Theorem 1.1 . We retain the selected embeddings and monomial weights and take D = 16 d 2 X D=16d^2X D = 16 d 2 X
Proposition 5.7 (Real-zero bound in the gamma height). Every real zero β ∈ ( 0 , 1 ) \beta\in(0,1) β ∈ ( 0 , 1 )
( 1 − β ) H K + ( 1 , 0 ) ≥ 1 1370 . (37) (1-\beta)\mathcal{H}^{+}_K(1,0)\ge\frac{1}{1370}. \tag*{(37)} ( 1 − β ) H K + ( 1 , 0 ) ≥ 1370 1 . (37) Proof. Choose
k = ⌈ 176 H log 2 ⌉ , X = 2 k , ℓ = log X , t = ℓ / H . k=\left\lceil\frac{176H}{\log2}\right\rceil,\qquad X=2^k,\qquad\ell=\log X,\qquad t=\ell/H. k = ⌈ log 2 176 H ⌉ , X = 2 k , ℓ = log X , t = ℓ / H . Since H ≥ n log ( 2 n ) + 4 n + 1 ≥ 5 H \ge n\log(2n) + 4n + 1 \ge5 H ≥ n log ( 2 n ) + 4 n + 1 ≥ 5
176 ≤ t ≤ 176 + log 2 H < 177 , ℓ ≥ 176 n log ( 2 n ) > 2 log d , ℓ > 2. 176 \le t \le176 + \frac{\log2}{H} < 177,\qquad\ell\ge176n\log(2n) > 2\log d,\qquad\ell> 2. 176 ≤ t ≤ 176 + H log 2 < 177 , ℓ ≥ 176 n log ( 2 n ) > 2 log d , ℓ > 2. Thus X > d 2 X > d^2 X > d 2 u = 2 u=2 u = 2 Lemma 5.5 . Lemmas 5.5 and 5.6 give
α 0 ℓ − 80 H ≤ e 2 Δ ℓ 2 2 + Δ ℓ ≤ 8 Δ ℓ 2 2 + Δ ℓ . \alpha_0\ell- 80H \le\frac{e^2\Delta\ell^2}{2+\Delta\ell} \le\frac{8\Delta\ell^2}{2+\Delta\ell}. α 0 ℓ − 80 H ≤ 2 + Δ ℓ e 2 Δ ℓ 2 ≤ 2 + Δ ℓ 8Δ ℓ 2 . Putting y = Δ H y=\Delta H y = Δ H α 0 t − 80 ≥ 85 > 0 \alpha_0t-80\ge85>0 α 0 t − 80 ≥ 85 > 0
y ≥ 2 ( α 0 t − 80 ) t { ( 8 − α 0 ) t + 80 } . (38) y \ge\frac{2(\alpha_0t-80)}{t\{(8-\alpha_0)t+80\}}. \tag*{(38)} y ≥ t {( 8 − α 0 ) t + 80 } 2 ( α 0 t − 80 ) . (38) For 176 ≤ t ≤ 177 176\le t\le177 176 ≤ t ≤ 177 1 / 1370 1/1370 1/1370
P ( t ) = − 113 t 2 + 39820 t − 3507200 ≥ 0. P(t)=-113t^2+39820t-3507200\ge0. P ( t ) = − 113 t 2 + 39820 t − 3507200 ≥ 0. Since P ′ ′ = − 226 P''=-226 P ′′ = − 226
P ( 176 ) = 832 , P ( 177 ) = 763 , P(176)=832,\qquad P(177)=763, P ( 176 ) = 832 , P ( 177 ) = 763 , concavity gives P ( t ) ≥ 763 > 0 P(t)\ge763>0 P ( t ) ≥ 763 > 0 [ 176 , 177 ] [176,177] [ 176 , 177 ]
Remark 5.8 (The cutoff choice). The choices u = 2 u=2 u = 2 176 ≤ ( log X ) / H < 177 176\le(\log X)/H<177 176 ≤ ( log X ) / H < 177 n log n n\log n n log n
Proof of Theorem 1.1 . For the principal character of K K K
H K ( 1 , 0 ) ≤ log D K + n log 4 + 4 n + 1 ≤ log D K + 6 n + 1 , H K + ( 1 , 0 ) ≤ B K . \mathcal{H}_K(1,0)\le\log D_K+n\log4+4n+1\le\log D_K+6n+1,\qquad\mathcal{H}_K^+(1,0)\le B_K. H K ( 1 , 0 ) ≤ log D K + n log 4 + 4 n + 1 ≤ log D K + 6 n + 1 , H K + ( 1 , 0 ) ≤ B K . Apply Proposition 5.7 .
Hecke zero-free regions
In this section, we deduce Hecke zero-free regions from the real-zero bound of § 5 and the positivity estimates of § 3 . We then give conductor formulations and extend the statements to imprimitive characters and real norm twists.
Quadratic characters and the general theorem
In this subsection, we transfer the real-zero bound of § 5.4 to quadratic Hecke characters by class-field factorization. Norm twists and the positivity estimates of § 3.2 then prove Theorem 1.3 .
Corollary 6.1 (Quadratic characters in the gamma height). For every primitive quadratic Hecke character of every number field,
L ( β , ξ ) = 0 , 0 < β < 1 ⟹ ( 1 − β ) H + ( ξ , 0 ) ≥ 1 4110 . L(\beta,\xi)=0,\qquad0<\beta<1\quad\Longrightarrow\quad(1-\beta)\mathcal{H}^+(\xi,0)\ge\frac{1}{4110}. L ( β , ξ ) = 0 , 0 < β < 1 ⟹ ( 1 − β ) H + ( ξ , 0 ) ≥ 4110 1 . Proof. Let F / K F/K F / K ξ \xi ξ [17 Chapter V, Theorem 3.27] gives d F / K = f ξ \mathfrak{d}_{F/K}=\mathfrak{f}_{\xi} d F / K = f ξ log D F = log D K + log A ξ \log D_F=\log D_K+\log A_{\xi} log D F = log D K + log A ξ ζ F = ζ K L ( s , ξ ) \zeta_F=\zeta_K L(s,\xi) ζ F = ζ K L ( s , ξ ) 0 , 0 0,0 0 , 0 0 , 1 0,1 0 , 1 0 , 1 0,1 0 , 1
H F ( 1 , 0 ) = H K ( 1 , 0 ) + H K ( ξ , 0 ) − 1 ≤ 2 H K ( ξ , 0 ) − 1. \mathcal{H}_F(1,0)=\mathcal{H}_K(1,0)+\mathcal{H}_K(\xi,0)-1\le2\mathcal{H}_K(\xi,0)-1. H F ( 1 , 0 ) = H K ( 1 , 0 ) + H K ( ξ , 0 ) − 1 ≤ 2 H K ( ξ , 0 ) − 1. The inequality uses H K ( 1 , 0 ) ≤ H K ( ξ , 0 ) \mathcal{H}_K(1,0)\le\mathcal{H}_K(\xi,0) H K ( 1 , 0 ) ≤ H K ( ξ , 0 ) ξ \xi ξ
H F + ( 1 , 0 ) ≤ 2 H K + ( ξ , 0 ) + 2 n log 2 ≤ 3 H K + ( ξ , 0 ) . \mathcal{H}^{+}_F(1,0)\le2\mathcal{H}^{+}_K(\xi,0)+2n\log2\le3\mathcal{H}^{+}_K(\xi,0). H F + ( 1 , 0 ) ≤ 2 H K + ( ξ , 0 ) + 2 n log 2 ≤ 3 H K + ( ξ , 0 ) . Apply Proposition 5.7 to the zero of ζ F \zeta_F ζ F
Proof of Theorem 1.3 . Since H + ( χ , t ) ≥ 5 \mathcal{H}^{+}(\chi,t)\ge5 H + ( χ , t ) ≥ 5 Re s > 0 \operatorname{Re}s>0 Re s > 0 § 3 therefore reduce the proof to a nontrivial zero β + i t \beta+it β + i t 0 < β < 1 0<\beta<1 0 < β < 1 χ \chi χ Proposition 3.3 . Otherwise write χ = ξ τ \chi=\xi\tau χ = ξ τ ξ \xi ξ u = t + τ u=t+\tau u = t + τ
L ( β + i t , χ ) = L ( β + i u , ξ ) , H + ( χ , t ) = H + ( ξ , u ) . L(\beta+it,\chi)=L(\beta+iu,\xi),\qquad\mathcal{H}^{+}(\chi,t)=\mathcal{H}^{+}(\xi,u). L ( β + i t , χ ) = L ( β + i u , ξ ) , H + ( χ , t ) = H + ( ξ , u ) . If u ≠ 0 u\ne0 u = 0 Proposition 3.4 applies. If u = 0 u=0 u = 0 Proposition 5.7 for principal ξ \xi ξ Corollary 6.1 for quadratic ξ \xi ξ ( 1 − β ) H + (1-\beta)\mathcal{H}^{+} ( 1 − β ) H +
1 112 , 1 1000 , 1 1370 , 1 4110 . \frac{1}{112},\qquad\frac{1}{1000},\qquad\frac{1}{1370},\qquad\frac{1}{4110}. 112 1 , 1000 1 , 1370 1 , 4110 1 . the smallest of which is 1 / 4110 1/4110 1/4110 Theorem 1.3 .
Finite-order characters and analytic conductors
In this subsection, we reformulate the theorem proved in § 6.1 for finite-order characters and analytic conductors. We use the conductor and gamma conventions of § 2.1 .
Corollary 6.2 (Finite-order characters). For a primitive finite-order Hecke character over a degree-n n n
σ > 1 − 1 4110 { log ( D K N f χ ) + n log ( 4 + ∣ t ∣ ) + n log ( 2 n ) + 4 n + 1 } . \sigma>1-\frac{1}{4110\{\log(D_KN\mathfrak{f}_{\chi})+n\log(4+|t|)+n\log(2n)+4n+1\}}. σ > 1 − 4110 { log ( D K N f χ ) + n log ( 4 + ∣ t ∣ ) + n log ( 2 n ) + 4 n + 1 } 1 . Principal poles are allowed. The constants are uniform as K K K χ \chi χ
Proof. The gamma parameters of a finite-order character are zero or one. Thus H + ( χ , t ) \mathcal{H}^{+}(\chi,t) H + ( χ , t ) Theorem 1.3 .
There is also an explicit coefficient at the height H \mathcal{H} H
L ( s , χ ) ≠ 0 if σ > 1 − 1 4110 { 1 + 1 5 log ( 2 n ) } H ( χ , t ) . (39) L(s,\chi)\ne0\ \text{if}\ \sigma>1-\frac{1}{4110\left\{1+\frac{1}{5}\log(2n)\right\}\mathcal{H}(\chi,t)}. \tag*{(39)} L ( s , χ ) = 0 if σ > 1 − 4110 { 1 + 5 1 log ( 2 n ) } H ( χ , t ) 1 . (39) Indeed H ( χ , t ) ≥ n log 3 + 4 n + 1 ≥ 5 n \mathcal{H}(\chi,t) \ge n\log3+4n+1 \ge5n H ( χ , t ) ≥ n log 3 + 4 n + 1 ≥ 5 n H + ≤ { 1 + 1 5 log ( 2 n ) } H \mathcal{H}^{+} \le\left\{1+\frac{1}{5}\log(2n)\right\}\mathcal{H} H + ≤ { 1 + 5 1 log ( 2 n ) } H (39) deteriorates logarithmically with n n n
In the range n log ( 2 n ) ≤ H ( χ , t ) n\log(2n) \le\mathcal{H}(\chi,t) n log ( 2 n ) ≤ H ( χ , t ) 1 / ( 8220 H ( χ , t ) ) 1/(8220\mathcal{H}(\chi,t)) 1/ ( 8220 H ( χ , t )) H \mathcal{H} H
For a standard analytic-conductor formulation put
C ( χ , t ) = D K N f χ ∏ j = 1 n ( 3 + ∣ μ j + i t ∣ ) . \mathfrak{C}(\chi,t) = D_K N\mathfrak{f}_{\chi} \prod_{j=1}^{n}(3+|\mu_j+it|). C ( χ , t ) = D K N f χ j = 1 ∏ n ( 3 + ∣ μ j + i t ∣ ) . Set C + ( χ , t ) = ( 2 n ) n C ( χ , t ) \mathfrak{C}^{+}(\chi,t)=(2n)^n\mathfrak{C}(\chi,t) C + ( χ , t ) = ( 2 n ) n C ( χ , t ) log C + ≥ n log 3 \log\mathfrak{C}^{+}\ge n\log3 log C + ≥ n log 3 4 n + 1 ≤ 5 n 4n+1\le5n 4 n + 1 ≤ 5 n H + ( χ , t ) ≤ 6 log C + ( χ , t ) \mathcal{H}^{+}(\chi,t)\le6\log\mathfrak{C}^{+}(\chi,t) H + ( χ , t ) ≤ 6 log C + ( χ , t )
L ( s , χ ) ≠ 0 if σ > 1 − 1 24660 log C + ( χ , t ) , (40) L(s,\chi)\ne0 \quad\text{if} \quad\sigma>1-\frac{1}{24660\log\mathfrak{C}^{+}(\chi,t)}, \tag*{(40)} L ( s , χ ) = 0 if σ > 1 − 24660 log C + ( χ , t ) 1 , (40) with the usual principal poles permitted.
Imprimitive characters and real norm twists
In this subsection, we extend the regions of § 6.1 –§ 6.2 to imprimitive characters and real norm twists. The norm convention of § 2.1 fixes the direction of translation.
For a deleted Euler factor of a unitary character,
1 − χ ( p ) ( N p ) − s = 0 , ∣ χ ( p ) ∣ = 1 ⟹ Re s = 0. 1-\chi(\mathfrak{p})(N\mathfrak{p})^{-s}=0,\quad|\chi(\mathfrak{p})|=1 \quad\Longrightarrow\quad\operatorname{Re}s=0. 1 − χ ( p ) ( N p ) − s = 0 , ∣ χ ( p ) ∣ = 1 ⟹ Re s = 0. Hence deleting Euler factors does not affect the stated regions in Re s > 0 \operatorname{Re}s>0 Re s > 0
For a continuous quasicharacter with primitive unitary part, write in the ideal-norm convention
ψ ( a ) = χ ( a ) ( N a ) − r , r ∈ R , χ primitive unitary . \psi(\mathfrak{a})=\chi(\mathfrak{a})(N\mathfrak{a})^{-r},\qquad r\in\mathbb{R},\quad\chi\text{ primitive unitary}. ψ ( a ) = χ ( a ) ( N a ) − r , r ∈ R , χ primitive unitary . Then
L ( s , ψ ) = L ( s + r , χ ) , Re s > 1 − r − 1 4110 H + ( χ , Im s ) ⟹ L ( s , ψ ) ≠ 0 , L(s,\psi)=L(s+r,\chi),\qquad\operatorname{Re}s>1-r-\frac{1}{4110\mathcal{H}^{+}(\chi,\operatorname{Im}s)} \quad\Longrightarrow\quad L(s,\psi)\ne0, L ( s , ψ ) = L ( s + r , χ ) , Re s > 1 − r − 4110 H + ( χ , Im s ) 1 ⟹ L ( s , ψ ) = 0 , with the translated principal poles permitted. For an imprimitive unitary part, apply this to its primitive inducing character and restore the deleted Euler factors. Their zeros translate from Re s = 0 \operatorname{Re}s=0 Re s = 0 Re s = − r \operatorname{Re}s=-r Re s = − r
Further comparison with Banks and earlier work
In this section, we compare the arithmetic argument of § 4 –§ 5 with Banks’ estimates and with earlier half-plane results. The comparisons distinguish the explicit field dependence of the present proof from stronger nonvanishing conclusions obtained by other methods.
What follows by normal closure
In this subsection, we justify the normal-closure deduction stated in § 1.3 and compare its degree dependence with Theorem 1.1 . Lemma 7.1 supplies the required discriminant estimate, including wild ramification.
Lemma 7.1 (Discriminant of a normal closure). Let E / Q E/\mathbb{Q} E / Q be a number field, let L L L be its normal closure, and put m = [ L : Q ] m=[L:\mathbb{Q}] m = [ L : Q ] Then
log D L ≤ m 2 log D E . (41) \log D_L \le\frac{m}{2}\log D_E. \tag*{(41)} log D L ≤ 2 m log D E . (41) This is a standard consequence of the conductor–discriminant formula; we include the short proof.
Proof. For a complex representation W W W J J J W J = { w ∈ W : g w = w for all g ∈ J } W^J=\{w\in W:gw=w\text{ for all }g\in J\} W J = { w ∈ W : g w = w for all g ∈ J } codim W J = dim W − dim W J \operatorname{codim}W^J=\dim W-\dim W^J codim W J = dim W − dim W J G = Gal ( L / Q ) G=\operatorname{Gal}(L/\mathbb{Q}) G = Gal ( L / Q ) V V V E E E L L L L L L J ⊂ G J\subset G J ⊂ G c J = dim V − dim V J c_J=\dim V-\dim V^J c J = dim V − dim V J c J = 1 c_J=1 c J = 1 ∣ J ∣ = 2 |J|=2 ∣ J ∣ = 2 c J ≥ 2 c_J\ge2 c J ≥ 2
1 − ∣ J ∣ − 1 ≤ 1 2 c J . 1-|J|^{-1}\le\frac{1}{2}c_J. 1 − ∣ J ∣ − 1 ≤ 2 1 c J . For the regular representation R G R_G R G
codim R G J = m ( 1 − ∣ J ∣ − 1 ) ≤ m 2 codim V J . \operatorname{codim}R_G^J=m(1-|J|^{-1})\le\frac{m}{2}\operatorname{codim}V^J. codim R G J = m ( 1 − ∣ J ∣ − 1 ) ≤ 2 m codim V J . It also holds for J = { 1 } J=\{1\} J = { 1 } p p p G i G_i G i G 0 G_0 G 0 a p ( W ) a_p(W) a p ( W ) p p p W W W G G G
a p ( W ) = ∑ i ≥ 0 ∣ G i ∣ ∣ G 0 ∣ codim W G i . a_p(W)=\sum_{i\ge0}\frac{|G_i|}{|G_0|}\operatorname{codim}W^{G_i}. a p ( W ) = i ≥ 0 ∑ ∣ G 0 ∣ ∣ G i ∣ codim W G i . Apply the preceding inequality to each G i G_i G i a p ( R G ) ≤ ( m / 2 ) a p ( V ) a_p(R_G)\le(m/2)a_p(V) a p ( R G ) ≤ ( m /2 ) a p ( V ) a p ( R G ) = v p ( D L ) a_p(R_G)=v_p(D_L) a p ( R G ) = v p ( D L ) a p ( V ) = v p ( D E ) a_p(V)=v_p(D_E) a p ( V ) = v p ( D E ) log p \log p log p
Proof of (9) . Let M M M K K K m = [ M : Q ] ≤ n ! m=[M:\mathbb{Q}]\le n! m = [ M : Q ] ≤ n ! M / K M/K M / K [4 ] makes ζ M / ζ K \zeta_M/\zeta_K ζ M / ζ K β ∈ ( 0 , 1 ) \beta\in(0,1) β ∈ ( 0 , 1 ) ζ K \zeta_K ζ K ζ M \zeta_M ζ M (8) and Lemma 7.1 :
c B ≤ ( 1 − β ) log D M ≤ m 2 ( 1 − β ) log D K . c_B\le(1-\beta)\log D_M\le\frac{m}{2}(1-\beta)\log D_K. c B ≤ ( 1 − β ) log D M ≤ 2 m ( 1 − β ) log D K . Consequently
( 1 − β ) log D K ≥ 2 c B m ≥ 2 c B n ! . (1-\beta)\log D_K\ge\frac{2c_B}{m}\ge\frac{2c_B}{n!}. ( 1 − β ) log D K ≥ m 2 c B ≥ n ! 2 c B . as asserted.
For K ≠ Q K\ne\mathbb{Q} K = Q
Normal-closure deduction: ( 1 − β ) log D K ≥ 2 c B / n ! , (1-\beta)\log D_K\ge2c_B/n!, ( 1 − β ) log D K ≥ 2 c B / n ! ,
Theorem 1.1: ( 1 − β ) log D K ≥ 1 1370 ( 1 + n log ( 2 n ) + 6 n + 1 log D K ) (1-\beta)\log D_K\ge\dfrac{1}{1370\left(1+\dfrac{n\log(2n)+6n+1}{\log D_K}\right)} ( 1 − β ) log D K ≥ 1370 ( 1 + log D K n log ( 2 n ) + 6 n + 1 ) 1
For normal or almost-normal K K K c / log D K c/\log D_K c / log D K K K K
Example 7.2. Let K = Q [ x ] / ( x 10 − x − 1 ) K = \mathbb{Q}[x]/(x^{10} - x - 1) K = Q [ x ] / ( x 10 − x − 1 ) M M M f = x 10 − x − 1 f = x^{10} - x - 1 f = x 10 − x − 1 Q \mathbb{Q} Q F 17 [ x ] \mathbb{F}_{17}[x] F 17 [ x ]
x 17 10 ≡ x ( m o d f ) , gcd ( f , x 17 2 − x ) = gcd ( f , x 17 5 − x ) = 1. x^{17^{10}} \equiv x \pmod{f}, \qquad\gcd(f,x^{17^2}-x)=\gcd(f,x^{17^5}-x)=1. x 1 7 10 ≡ x ( mod f ) , g cd( f , x 1 7 2 − x ) = g cd( f , x 1 7 5 − x ) = 1. These are the irreducibility criterion in degree 10, whose prime divisors are 2 and 5. The polynomial has discriminant 10 10 + 9 9 = 10387420489 = 173 ⋅ 60042893 10^{10}+9^9=10387420489=173\cdot60042893 1 0 10 + 9 9 = 10387420489 = 173 ⋅ 60042893 D K D_K D K [23 Corollary 3] on x n − x − 1 x^n-x-1 x n − x − 1 S 10 S_{10} S 10
[ K : Q ] = 10 , D K = 10387420489 , Gal ( M / Q ) ≃ S 10 , [ M : Q ] = 10 ! . [K:\mathbb{Q}]=10,\qquad D_K=10387420489,\qquad\operatorname{Gal}(M/\mathbb{Q})\simeq S_{10},\qquad[M:\mathbb{Q}]=10!. [ K : Q ] = 10 , D K = 10387420489 , Gal ( M / Q ) ≃ S 10 , [ M : Q ] = 10 ! . Theorem 1.1 gives the real-zero gap
w a r i t h = 1 1370 { log D K + 10 log 20 + 61 } ≈ 6.40168 × 10 − 6 . w_{\mathrm{arith}}=\frac{1}{1370\{\log D_K+10\log20+61\}}\approx6.40168\times10^{-6}. w arith = 1370 { log D K + 10 log 20 + 61 } 1 ≈ 6.40168 × 1 0 − 6 . For the admissible normal-field constant furnished by [1 equation (3.3)] , one has c B ≤ 1 / 4 c_B\leq1/4 c B ≤ 1/4
w c l = 2 c B 10 ! log D K ≤ 1 2 ⋅ 10 ! log D K ≈ 5.97413 × 10 − 9 . w_{\mathrm{cl}}=\frac{2c_B}{10!\log D_K}\leq\frac{1}{2\cdot10!\log D_K}\approx5.97413\times10^{-9}. w cl = 10 ! log D K 2 c B ≤ 2 ⋅ 10 ! log D K 1 ≈ 5.97413 × 1 0 − 9 . For the same c B c_B c B M M M
w B a n k s = c B log D M . w_{\mathrm{Banks}}=\frac{c_B}{\log D_M}. w Banks = log D M c B . The discriminant tower formula for M / K M/K M / K
D M = D K [ M : K ] N K / Q d M / K ≥ D K 10 ! / 10 , D_M=D_K^{[M:K]}N_{K/\mathbb{Q}}\mathfrak{d}_{M/K}\geq D_K^{10!/10}, D M = D K [ M : K ] N K / Q d M / K ≥ D K 10 ! /10 , where d M / K \mathfrak{d}_{M/K} d M / K
w B a n k s ≤ 10 c B 10 ! log D K = 5 w c l . w_{\mathrm{Banks}}\leq\frac{10c_B}{10!\log D_K}=5w_{\mathrm{cl}}. w Banks ≤ 10 ! log D K 10 c B = 5 w cl . Consequently,
w a r i t h > 1071 w c l ≥ 1071 5 w B a n k s = 214.2 w B a n k s . w_{\mathrm{arith}}>1071w_{\mathrm{cl}}\geq\frac{1071}{5}w_{\mathrm{Banks}}=214.2w_{\mathrm{Banks}}. w arith > 1071 w cl ≥ 5 1071 w Banks = 214.2 w Banks . In particular, w a r i t h > 214.2 w B a n k s w_{\mathrm{arith}}>214.2w_{\mathrm{Banks}} w arith > 214.2 w Banks M M M c B c_B c B c B ( K ) c_B(K) c B ( K )
The arithmetic estimates
In this subsection, we compare Banks’ prime-sum estimate [1 Theorem 1.5] with the determinant construction of § 4 and its specialization in § 5.1 . We then distinguish the normal-closure deduction of § 7.1 from Banks’ direct results and give a quadratic example in which Banks’ explicit bound is stronger.
With L / k L/k L / k H 0 H_0 H 0 K / k K/k K / k H H H L / k L/k L / k G G G 1 ∈ S ⊂ G 1\in S\subset G 1 ∈ S ⊂ G λ L \lambda_L λ L L L L p \mathfrak{p} p k k k L L L δ p = ∣ S ∩ C p ∣ / ∣ C p ∣ \delta_p = |S \cap C_p|/|C_p| δ p = ∣ S ∩ C p ∣/∣ C p ∣ C p C_p C p G G G δ p = 0 \delta_p = 0 δ p = 0 H 0 ≥ 2 H_0 \ge2 H 0 ≥ 2
∑ N p ≤ X δ p log N p N p ≤ ( 1 + ∣ S ∣ − 1 H 0 ) ( ∣ S ∣ ∣ G ∣ log X + [ k : Q ] log λ L ) + O [ L : Q ] , H 0 ( 1 ) , (42) \sum_{N\mathfrak p\le X}\delta_p\frac{\log Np}{Np}\leq\left(1+\frac{|S|-1}{H_0}\right)\left(\frac{|S|}{|G|}\log X+[k:\mathbb{Q}]\log\lambda_L\right)+O_{[L:\mathbb{Q}],H_0}(1), \tag*{(42)} N p ≤ X ∑ δ p N p log N p ≤ ( 1 + H 0 ∣ S ∣ − 1 ) ( ∣ G ∣ ∣ S ∣ log X + [ k : Q ] log λ L ) + O [ L : Q ] , H 0 ( 1 ) , (42) for
X ≥ max { X 0 ( [ L : Q ] , H 0 ) , ( log D L ) [ L : Q ] } . X\geq\max\{X_0([L:\mathbb{Q}],H_0),(\log D_L)^{[L:\mathbb{Q}]}\}. X ≥ max { X 0 ([ L : Q ] , H 0 ) , ( log D L ) [ L : Q ] } . Here X 0 ( [ L : Q ] , H 0 ) X_0([L:\mathbb{Q}],H_0) X 0 ([ L : Q ] , H 0 )
Here Lemma 4.4 uses small independent integers of E E E M M M λ E ≤ D E 2 / d \lambda_E\leq D_E^{2/d} λ E ≤ D E 2/ d H 0 = 16 d 2 H_0=16d^2 H 0 = 16 d 2 D = 16 d 2 X D=16d^2X D = 16 d 2 X
∑ p < X δ p log p p ≤ ( 1 − 15 16 d ) log X + 4 d log D E + 32 log ( 2 d ) , X ∈ Z , X ≥ d 2 . \sum_{p<X}\delta_p\frac{\log p}{p}\leq\left(1-\frac{15}{16d}\right)\log X+\frac{4}{d}\log D_E+32\log(2d),\qquad X\in\mathbb{Z},\quad X\geq d^2. p < X ∑ δ p p log p ≤ ( 1 − 16 d 15 ) log X + d 4 log D E + 32 log ( 2 d ) , X ∈ Z , X ≥ d 2 . The lower comparison in § 5 uses 1 − δ p ≤ r K ( p ) / n 1-\delta_p\leq r_K(p)/n 1 − δ p ≤ r K ( p ) / n p ∤ D E p\nmid D_E p ∤ D E K K K
These estimates display the dependence on the varying field. The determinant comparison follows [20 , 2 ] ; its injectivity step uses Masser’s theorem as in [9 ] and the selected-embedding calculation of [1 ] .
The normal-closure comparison of § 7.1 uses two successive lower bounds. For K ≠ Q K\neq\mathbb{Q} K = Q M M M m = [ M : Q ] m=[M:\mathbb{Q}] m = [ M : Q ]
1 − β ≥ c B log D M ≥ 2 c B m log D K . 1-\beta\geq\frac{c_B}{\log D_M}\geq\frac{2c_B}{m\log D_K}. 1 − β ≥ log D M c B ≥ m log D K 2 c B . The width w c l w_{\mathrm{cl}} w cl D M D_M D M w B a n k s ≤ 5 w c l w_{\mathrm{Banks}}\leq5w_{\mathrm{cl}} w Banks ≤ 5 w cl w a r i t h > 214.2 w B a n k s w_{\mathrm{arith}}>214.2\,w_{\mathrm{Banks}} w arith > 214.2 w Banks c B c_B c B
Banks also gives direct real-zero bounds for quadratic extensions F / k F/k F / k L / k L/k L / k k k k [1 Corollary 1.2 and Theorem 1.3] , with effective constants depending on k k k Q \mathbb{Q} Q L / Q L/\mathbb{Q} L / Q (8) gives
1 − β ≥ c B log D L , 1-\beta\geq\frac{c_B}{\log D_L}, 1 − β ≥ log D L c B , without the additional degree term in Theorem 1.1. For almost-normal fields, see [1 Remark 3.4] . A numerical comparison requires the values of the relevant constants; Banks’ earlier rational result gives the following explicit example.
Example 7.3 (A quadratic family with a stronger Banks bound). Let χ \chi χ q ≥ 10 10 q\geq10^{10} q ≥ 1 0 10 F F F D F = q D_F=q D F = q ζ F = ζ Q L ( s , χ ) \zeta_F=\zeta_{\mathbb{Q}}L(s,\chi) ζ F = ζ Q L ( s , χ ) ζ Q \zeta_{\mathbb{Q}} ζ Q ( 0 , 1 ) (0,1) ( 0 , 1 ) β ∈ ( 0 , 1 ) \beta\in(0,1) β ∈ ( 0 , 1 ) ζ F \zeta_F ζ F L ( s , χ ) L(s,\chi) L ( s , χ ) [2 Theorem 1.1] and Theorem 1.1 give, respectively,
w B a n k s = 3 1000 log q , w a r i t h = 1 1370 { log q + 2 log 4 + 13 } . w_{\mathrm{Banks}}=\frac{3}{1000\log q},\qquad w_{\mathrm{arith}}=\frac{1}{1370\{\log q+2\log4+13\}}. w Banks = 1000 log q 3 , w arith = 1370 { log q + 2 log 4 + 13 } 1 . Thus
w B a n k s w a r i t h = 411 100 ( 1 + 2 log 4 + 13 log q ) > 411 100 = 4.11. \frac{w_{\mathrm{Banks}}}{w_{\mathrm{arith}}}=\frac{411}{100}\left(1+\frac{2\log4+13}{\log q}\right)>\frac{411}{100}=4.11. w arith w Banks = 100 411 ( 1 + log q 2 log 4 + 13 ) > 100 411 = 4.11. Banks’ bound is therefore numerically stronger in this family. With the finite computation cited in [2 Theorem 1.1] , the same comparison holds for every q ≥ 3 q \ge3 q ≥ 3
Earlier half-plane results
In this subsection, we recall earlier half-plane results and compare their consequences with the regions of § 6 . These results are not used in the arithmetic proof of § 4 –§ 5 . For the historical discussion, see [13 Section 1.4] .
OpenAI’s cubic-theta argument [19 ] gives
L F ( s , η ) ≠ 0 ( Re s > 7 / 8 ) , F = Q ( − 3 ) , L_F(s,\eta) \ne0 \quad(\operatorname{Re}s > 7/8), \qquad F=\mathbb{Q}(\sqrt{-3}), L F ( s , η ) = 0 ( Re s > 7/8 ) , F = Q ( − 3 ) , for every finite-order Hecke character η \eta η F F F L L L [21 ] gives the boundary 11 / 12 11/12 11/12 [20 ] .
Varying the length parameters in the cubic-theta comparison, the first version of [11 ] gives
B r = 34999 40000 = 0.874975 , B a l g = 1507 − 2 921 1653 = 0.874957069799 … . B_{\mathrm{r}}=\frac{34999}{40000}=0.874975,\qquad B_{\mathrm{alg}}=\frac{1507-2\sqrt{921}}{1653}=0.874957069799\ldots. B r = 40000 34999 = 0.874975 , B alg = 1653 1507 − 2 921 = 0.874957069799 … . Mermet [16 ] and Wei [26 ] report 0.8749570698. Chakhkiev [5 ] reports the conditional boundary 10937 / 12500 = 0.87496 10937/12500=0.87496 10937/12500 = 0.87496 [11 Theorem 27.1] gives
B n e w = 0.874957019420098946 … . B_{\mathrm{new}}=0.874957019420098946\ldots. B new = 0.874957019420098946 … . Xu and Sang [27 ] use an amplified average for the reflected numerator, obtaining B X S = 8748867 / 10 7 = 0.8748867 B_{\mathrm{XS}}=8748867/10^{7}=0.8748867 B XS = 8748867/1 0 7 = 0.8748867 [19 ] . Newton [18 Introduction, pp. 1–4] gives the certified comparison boundary B N = 43744332481 / 50000000000 = 0.87488664962 B_{\mathrm{N}}=43744332481/50000000000=0.87488664962 B N = 43744332481/50000000000 = 0.87488664962
For arbitrary number fields, Cureton–Thorner [7 Theorem 1.1 and Corollary 1.2] give nonvanishing in Re s > 31 / 32 \operatorname{Re}s > 31/32 Re s > 31/32 L L L [27 ] and the general-field argument of [7 ] , the author [12 Theorem 17.1] obtains the uniform boundary
B ♭ = 0.874886647039964 … , 133488 B ♭ 4 − 376596 B ♭ 3 + 384534 B ♭ 2 − 167541 B ♭ + 26231 = 0. B_{\flat}=0.874886647039964\ldots,\qquad133488B_{\flat}^{4}-376596B_{\flat}^{3}+384534B_{\flat}^{2}-167541B_{\flat}+26231=0. B ♭ = 0.874886647039964 … , 133488 B ♭ 4 − 376596 B ♭ 3 + 384534 B ♭ 2 − 167541 B ♭ + 26231 = 0. The general-field metaplectic argument is carried out over K ( ζ 12 ) K(\zeta_{12}) K ( ζ 12 ) ζ 12 = e 2 π i / 12 \zeta_{12}=e^{2\pi i/12} ζ 12 = e 2 π i /12 K K K
The subsequent refinement [13 Theorem 1.1] gives, for every finite-order Hecke character over every number field,
L K ( s , χ ) ≠ 0 ( Re s > B t w ) , B t w = 0.87405850379185900499 … (43) L_K(s,\chi)\ne0\quad(\operatorname{Re}s>B_{\mathrm{tw}}),\qquad B_{\mathrm{tw}}=0.87405850379185900499\ldots \tag*{(43)} L K ( s , χ ) = 0 ( Re s > B tw ) , B tw = 0.87405850379185900499 … (43) with the principal pole permitted. An amplified Gauss norm and contour displacement, with an iteration of the character-counting estimate, first give B ∗ = 0.8744683076933045955 … B_{\ast}=0.8744683076933045955\ldots B ∗ = 0.8744683076933045955 … B e x = 0.87407284766682790777 … B_{\mathrm{ex}} = 0.87407284766682790777\ldots B ex = 0.87407284766682790777 … [13 Theorem 1.2] . Selected unit primes entering the Gauss sums as a moving cubic character then give B t w B_{\mathrm{tw}} B tw
The selected character-sum estimates and finite moment inductions of Bebel, ArviFoxy, Sil and collaborators are extended from Dirichlet characters to ideal characters in [14 ] (for these sources and the same limiting boundary in the Dirichlet family, see [14 Section 1.3] ). Theorem 1.1 there gives, for every finite-order Hecke character over every number field,
L K ( s , χ ) ≠ 0 ( Re s > B 2 ) , B 2 = 777 − 1081 852 = 0.873381966613207519929 … , (44) L_K(s,\chi) \ne0 \quad(\operatorname{Re} s > B_2), \qquad B_2 = \frac{777-\sqrt{1081}}{852}=0.873381966613207519929\ldots, \tag*{(44)} L K ( s , χ ) = 0 ( Re s > B 2 ) , B 2 = 852 777 − 1081 = 0.873381966613207519929 … , (44) with the principal pole permitted; here 1278 B 2 2 − 2331 B 2 + 1061 = 0 1278B_2^2-2331B_2+1061=0 1278 B 2 2 − 2331 B 2 + 1061 = 0 [14 Theorem 4.9] for smooth sums S n 1 , 1 , S n 2 , 2 S_{n_1,1},S_{n_2,2} S n 1 , 1 , S n 2 , 2 Q Q Q z z z d = n 1 + n 2 d=n_1+n_2 d = n 1 + n 2
d + 6 κ 1 z ≤ M ⟶ d + 3 z ≤ M ⟶ d + 2 z ≤ M , κ 1 = 0.749774. d+6\kappa_1z\le M \quad\longrightarrow\quad d+3z\le M \quad\longrightarrow\quad d+2z\le M,\qquad\kappa_1=0.749774. d + 6 κ 1 z ≤ M ⟶ d + 3 z ≤ M ⟶ d + 2 z ≤ M , κ 1 = 0.749774. The larger prime length improves both the zero-detector count and the averaged numerator of the cubic-theta comparison, and displacing the numerator contour adds a second saving. By [14 Theorem 6.9] , B 2 B_2 B 2
B 2 < B t w < B ∗ < B ♭ < 31 32 . B_2<B_{\mathrm{tw}}<B_*<B_{\flat}<\frac{31}{32}. B 2 < B tw < B ∗ < B ♭ < 32 31 . For completeness, even the boundary 31 / 32 31/32 31/32 Theorem 1.3 . Indeed, for a principal or quadratic character ξ \xi ξ β + i u \beta+iu β + i u
1 − β ≥ 1 32 , ( 1 − β ) H ( ξ , u ) ≥ 5 32 , 1-\beta\ge\frac{1}{32},\qquad(1-\beta)\mathcal{H}(\xi,u)\ge\frac{5}{32}, 1 − β ≥ 32 1 , ( 1 − β ) H ( ξ , u ) ≥ 32 5 , since H ≥ 5 \mathcal{H}\ge5 H ≥ 5 Proposition 3.3 gives, for every primitive unitary Hecke character,
L ( s , χ ) ≠ 0 if σ > 1 − 1 112 H ( χ , t ) , L(s,\chi)\ne0 \quad\text{if }\sigma>1-\frac{1}{112\mathcal{H}(\chi,t)}, L ( s , χ ) = 0 if σ > 1 − 112 H ( χ , t ) 1 , with the usual principal poles permitted. The boundaries B ♭ B_{\flat} B ♭ B ∗ B_* B ∗ B t w B_{\mathrm{tw}} B tw B 2 B_2 B 2
Theorems 1.1 and 1.3 follow here from the arithmetic estimates of § 4 –§ 5 and classical positivity. Their nonvanishing conclusions also follow from the preceding half-plane results together with classical positivity.
Numerical inequalities
In this appendix, we collect the numerical inequalities used in § 1.1 , § 5.1 , § 5.4 , and § 6.1 . All inequalities hold in every degree, including the smallest permitted parameter values.
The degree and discriminant. Minkowski’s bound implies
D K ≥ ( π 4 ) n ( n n n ! ) 2 . D_K\ge\left(\frac{\pi}{4}\right)^n\left(\frac{n^n}{n!}\right)^2. D K ≥ ( 4 π ) n ( n ! n n ) 2 . For a n = ( π / 4 ) n ( n n / n ! ) 2 e − n / 2 a_n=(\pi/4)^n(n^n/n!)^2e^{-n/2} a n = ( π /4 ) n ( n n / n ! ) 2 e − n /2 a 3 > 1 a_3>1 a 3 > 1
a n + 1 a n = π 4 e ( 1 + 1 n ) 2 n ≥ π e > 1. \frac{a_{n+1}}{a_n}=\frac{\pi}{4\sqrt{e}}\left(1+\frac{1}{n}\right)^{2n}\ge\frac{\pi}{\sqrt{e}}>1. a n a n + 1 = 4 e π ( 1 + n 1 ) 2 n ≥ e π > 1. Thus log D K ≥ n / 2 \log D_K\ge n/2 log D K ≥ n /2 n ≥ 3 n\ge3 n ≥ 3 n = 2 n=2 n = 2 D K ≥ 3 D_K\ge3 D K ≥ 3 log D K > 1 \log D_K>1 log D K > 1 K ≠ Q K\ne\mathbb{Q} K = Q
B K log D K ≤ 14 + 2 log ( 2 n ) ≤ 16 log ( 2 n ) , \frac{B_K}{\log D_K}\le14+2\log(2n)\le16\log(2n), log D K B K ≤ 14 + 2 log ( 2 n ) ≤ 16 log ( 2 n ) , which proves (2) .
The polynomial degree and prime cutoff. For D ≥ 1 D\ge1 D ≥ 1 log ( D + 1 ) ≤ D \log(D+1)\le\sqrt{D} log ( D + 1 ) ≤ D D = 1 D=1 D = 1 ( D − 1 ) 2 / ( 2 D ( D + 1 ) ) ≥ 0 (\sqrt{D}-1)^2/(2\sqrt{D}(D+1))\ge0 ( D − 1 ) 2 / ( 2 D ( D + 1 )) ≥ 0 d ≥ 2 d\ge2 d ≥ 2 s = d − 1 s=d-1 s = d − 1 H 0 = 16 d 2 H_0=16d^2 H 0 = 16 d 2 D = 16 d 2 X D=16d^2X D = 16 d 2 X X ≥ d 2 X\ge d^2 X ≥ d 2 D ≥ 16 d 4 ≥ 2 H 0 D\ge16d^4\ge2H_0 D ≥ 16 d 4 ≥ 2 H 0
( 1 + d − 2 16 d 2 ) d − 1 d ≤ 1 − 15 16 d 2 , 2 d ϑ ( X ) D ≤ 1 2 d , d ( d − 1 ) D log ( D + 1 ) ≤ 1 4 . \left(1+\frac{d-2}{16d^2}\right)\frac{d-1}{d}\le1-\frac{15}{16d^2},\qquad\frac{2d\vartheta(X)}{D}\le\frac{1}{2d},\qquad\frac{d(d-1)}{D}\log(D+1)\le\frac{1}{4}. ( 1 + 16 d 2 d − 2 ) d d − 1 ≤ 1 − 16 d 2 15 , D 2 d ϑ ( X ) ≤ 2 d 1 , D d ( d − 1 ) log ( D + 1 ) ≤ 4 1 . The last inequality follows from D ≥ 4 d 2 \sqrt{D}\ge4d^2 D ≥ 4 d 2
5 + 23 log 2 + 14 log d ≤ 24 + 14 log d ≤ 32 log ( 2 d ) . 5+23\log2+14\log d\le24+14\log d\le32\log(2d). 5 + 23 log 2 + 14 log d ≤ 24 + 14 log d ≤ 32 log ( 2 d ) . The last step follows from d ≥ 2 d\ge2 d ≥ 2 log 2 > 1 / 2 \log2>1/2 log 2 > 1/2
The error bound and integer cutoff. Put H = H K + ( 1 , 0 ) H=\mathcal{H}^{+}_K(1,0) H = H K + ( 1 , 0 ) E E E d d d B E B_E B E (29) . The bound log ( 2 + B E ) ≤ log ( 2 d ) + B E / d \log(2+B_E)\le\log(2d)+B_E/d log ( 2 + B E ) ≤ log ( 2 d ) + B E / d e 2 < 8 e^2<8 e 2 < 8 E E ( 2 ) ≤ 80 H \mathcal{E}_E(2)\le80H E E ( 2 ) ≤ 80 H k = ⌈ 176 H / log 2 ⌉ k=\lceil176H/\log2\rceil k = ⌈ 176 H / log 2 ⌉ X = 2 k X=2^k X = 2 k
176 ≤ t = ( log X ) / H < 177 , X > d 2 , log X > 2. 176\le t=(\log X)/H<177,\qquad X>d^2,\qquad\log X>2. 176 ≤ t = ( log X ) / H < 177 , X > d 2 , log X > 2. The exact polynomial inequality in Proposition 5.7 is
P ( t ) = − 113 t 2 + 39820 t − 3507200 , P ( 176 ) = 832 , P ( 177 ) = 763 , P ′ ′ ( t ) = − 226. P(t)=-113t^2+39820t-3507200,\qquad P(176)=832,\qquad P(177)=763,\qquad P''(t)=-226. P ( t ) = − 113 t 2 + 39820 t − 3507200 , P ( 176 ) = 832 , P ( 177 ) = 763 , P ′′ ( t ) = − 226. Concavity gives P ( t ) ≥ 763 > 0 P(t)\ge763>0 P ( t ) ≥ 763 > 0 [ 176 , 177 ] [176,177] [ 176 , 177 ] ( 1 − β ) H ≥ 1 / 1370 (1-\beta)H\ge1/1370 ( 1 − β ) H ≥ 1/1370
For a quadratic class field F / K F/K F / K
H F + ( 1 , 0 ) ≤ 2 H K + ( ξ , 0 ) + 2 n log 2 − 1 ≤ 3 H K + ( ξ , 0 ) . \mathcal{H}^{+}_F(1,0)\le2\mathcal{H}^{+}_K(\xi,0)+2n\log2-1\le3\mathcal{H}^{+}_K(\xi,0). H F + ( 1 , 0 ) ≤ 2 H K + ( ξ , 0 ) + 2 n log 2 − 1 ≤ 3 H K + ( ξ , 0 ) . Since 3 ⋅ 1370 = 4110 3\cdot1370=4110 3 ⋅ 1370 = 4110 H \mathcal{H} H 2 ⋅ 4110 = 8220 2\cdot4110=8220 2 ⋅ 4110 = 8220 6 ⋅ 4110 = 24660 6\cdot4110=24660 6 ⋅ 4110 = 24660
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
liu2053@purdue.edu
