Upper bounds on gaps between zeros of $L$-functions

Tianyu Zhao

arxiv: 2511.13898 · v2 · pith:6TCEJPKUnew · submitted 2025-11-17 · 🧮 math.NT

Upper bounds on gaps between zeros of L-functions

Tianyu Zhao This is my paper

Pith reviewed 2026-05-21 17:56 UTC · model grok-4.3

classification 🧮 math.NT

keywords L-functionsnon-trivial zerosgaps between zerosupper boundsanalytic conductorcontour integrationfunctional equation

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The pith

Unconditional upper bounds hold for gaps between consecutive non-trivial zeros of any general L-function.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves two unconditional upper bounds on gaps between consecutive non-trivial zero ordinates for general L-functions. These extend Hall-Hayman for the zeta function and Siegel for Dirichlet L-functions using contour integration. The bounds hold under standard analytic properties without extra assumptions. Readers care because zero spacings inform prime distributions via explicit formulae. One method is sharper when degree is small relative to analytic conductor, the other otherwise.

Core claim

We prove two unconditional upper bounds on the gaps between ordinates of consecutive non-trivial zeros of a general L-function L(s). This extends previous work of Hall and Hayman (2000) on the Riemann zeta-function and work of Siegel (1945) on Dirichlet L-functions. We observe that Hall and Hayman's method gives a sharper estimate when the degree of L(s) is sufficiently small compared to the analytic conductor, while Siegel's method performs better in the other regime.

What carries the argument

Direct application of the Hall-Hayman and Siegel contour-integration arguments to general L-functions via analytic continuation and functional equation.

If this is right

The gap is bounded by an explicit function of the analytic conductor.
Hall-Hayman gives the sharper bound for small degree relative to conductor.
Siegel gives the sharper bound in the complementary regime.
The results require no hypotheses beyond the standard analytic properties of L-functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The bounds could supply quantitative inputs for prime gap estimates in arithmetic progressions.
A hybrid of the two methods might produce a uniform bound across all degree-conductor regimes.
Similar contour techniques may extend to zeros of L-functions attached to higher-rank automorphic forms.

Load-bearing premise

The L-function must satisfy analytic continuation to the whole plane, a functional equation, and growth conditions permitting direct use of the contour-integration methods.

What would settle it

An explicit computation for some L-function where the gap between two consecutive non-trivial zero ordinates exceeds both derived upper bounds.

read the original abstract

We prove two unconditional upper bounds on the gaps between ordinates of consecutive non-trivial zeros of a general $L$-function $L(s)$. This extends previous work of Hall and Hayman (2000) on the Riemann zeta-function and work of Siegel (1945) on Dirichlet $L$-functions. Interestingly, we observe that while Hall and Hayman's method gives a sharper estimate when the degree of $L(s)$ is sufficiently small compared to the analytic conductor, Siegel's method does better in the other regime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Zhao extends Hall-Hayman and Siegel to give two unconditional gap bounds for general L-functions, with a useful regime comparison between the methods.

read the letter

The main thing here is that Zhao has extended the Hall-Hayman contour method and the Siegel argument to produce two explicit unconditional upper bounds on gaps between consecutive non-trivial zeros of a general L-function, plus a direct comparison showing which bound is better when the degree is small relative to the analytic conductor and vice versa. The work assumes only the standard analytic continuation, functional equation, and growth conditions that let the classical estimates apply directly. What is new is the uniform treatment across arbitrary degree and conductor together with the regime split; the core techniques are the old ones, but putting them side by side for the general case and noting the crossover is a clear addition. The paper does a straightforward job of carrying the logarithmic-derivative estimates and Gamma-factor contributions over without introducing extra hypotheses or fitting parameters to the target gap. That keeps the claims clean and unconditional. The soft spots are limited. The bounds rest on whether the error terms from the contour shifts and residue calculations remain controlled once the degree and conductor grow; the abstract asserts they do, but without the full derivation or explicit constants it is hard to judge how much room is left in the final inequalities. If those steps are just the standard ones with parameters tracked carefully, the argument should hold. If any hidden dependence on degree appears in the Gamma estimates, the uniformity could weaken. This is the sort of paper for analytic number theorists who already use zero-spacing results for zeta or Dirichlet L-functions and want versions that apply to higher-degree or higher-conductor families. A reader working on explicit zero-density or spacing applications would find the regime comparison practical. It deserves a serious referee because the claims are concrete, the methods are standard but applied systematically, and the central extension looks verifiable from the stated assumptions. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The manuscript proves two unconditional upper bounds on the gaps between ordinates of consecutive non-trivial zeros of a general L-function L(s). It extends the contour-integration argument of Hall and Hayman (2000) for the Riemann zeta-function and the argument of Siegel (1945) for Dirichlet L-functions. The authors observe a regime-dependent comparison: Hall-Hayman yields the sharper bound when the degree is small relative to the analytic conductor, while Siegel's method is superior in the complementary regime.

Significance. If the derivations hold, the results extend classical zero-gap bounds to L-functions of arbitrary degree under only the standard analytic continuation, functional equation, and growth hypotheses. The explicit regime comparison between the two methods is a useful contribution that clarifies their relative strengths. The unconditional character of the proofs, without additional hypotheses or parameter fitting, is a clear strength.

major comments (2)

[§2] §2 (Hall-Hayman adaptation): the text must explicitly track how the contribution of the Gamma factors in the functional equation and the bound on L'/L scale with the degree d and analytic conductor Q; without this dependence made uniform, the claimed gap bound does not follow directly from the classical estimates.
[§3] §3 (Siegel adaptation): the residue estimate after the contour shift requires verification that the zero-free region or the relevant growth bounds remain valid for general degree; the manuscript should state the precise form of the resulting gap bound in terms of d and Q.

minor comments (2)

The introduction should include a precise definition of the class of L-functions to which the results apply (e.g., the Selberg class axioms or equivalent).
Notation for the analytic conductor should be introduced once and used consistently in all statements of the main theorems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. The points raised concern the need for explicit uniformity in the degree d and analytic conductor Q. We have revised the manuscript to address both comments directly by adding the required tracking and verifications.

read point-by-point responses

Referee: [§2] §2 (Hall-Hayman adaptation): the text must explicitly track how the contribution of the Gamma factors in the functional equation and the bound on L'/L scale with the degree d and analytic conductor Q; without this dependence made uniform, the claimed gap bound does not follow directly from the classical estimates.

Authors: We agree that the dependence must be tracked explicitly for the argument to be uniform. In the revised manuscript we have inserted a dedicated paragraph in §2 that applies Stirling's formula to the Gamma factors in the functional equation, yielding a contribution bounded by O(d log(Q(1 + |t|))). We likewise record a uniform bound for |L'/L| on the relevant vertical and horizontal segments, obtained from the standard convexity estimates that hold for general L-functions under the stated hypotheses. These additions make the contour-integration argument self-contained and confirm that the claimed gap bound follows directly. revision: yes
Referee: [§3] §3 (Siegel adaptation): the residue estimate after the contour shift requires verification that the zero-free region or the relevant growth bounds remain valid for general degree; the manuscript should state the precise form of the resulting gap bound in terms of d and Q.

Authors: The referee is correct that an explicit verification is needed. The zero-free region employed is the classical one derived from the functional equation and Phragmén-Lindelöf convexity, which remains valid for L-functions of arbitrary degree d with analytic conductor Q (with the implied constant allowed to depend on d). We have added a short lemma in §3 confirming the requisite growth bounds on the shifted contour. The resulting gap bound is now stated explicitly in terms of d and Q at the end of the section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via classical estimates

full rationale

The paper adapts the Hall-Hayman contour-integration argument and the Siegel argument to general L-functions under the explicit assumption of standard analytic continuation, functional equation, and growth bounds. These are independent external hypotheses, not derived from or fitted to the target zero-gap bounds. No parameters are tuned to the output gaps, no self-citations form the load-bearing justification, and the regime comparison (degree versus analytic conductor) follows directly from comparing the resulting explicit estimates without renaming or re-deriving the inputs. The work is therefore a straightforward extension of prior independent results rather than a closed loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard analytic properties of L-functions and the applicability of two classical contour-integration techniques; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption L-functions admit analytic continuation to the whole plane, satisfy a functional equation, and obey standard growth estimates in the critical strip.
Invoked implicitly to justify application of Hall-Hayman and Siegel methods to a general L-function.

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discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Counting zeros of DirichletL-functions

[Ben+21] M. A. Bennett, G. Martin, K. O’Bryant, and A. Rechnitzer. “Counting zeros of DirichletL-functions”. In:Math. Comp.90(329) (2021), pp. 1455–1482. [Bob+15] J. Bober, J. B. Conrey, D. W. Farmer, A. Fujii, S. Koutsoliotas, S. Lemurell, M. Rubinstein, and H. Yoshida. “The highest lowest zero of generalL-functions”. In:J. Number Theory147 (2015), pp. 3...

work page 2021
[2]

Two notes on the Riemann zeta-function

[Lit24] J. E. Littlewood. “Two notes on the Riemann zeta-function”. In:Proc. Cam- bridge Philos. Soc.22 (1924), pp. 234–242. [Mil02] S. D. Miller. “The highest lowest zero and other applications of positivity”. In: Duke Math. J.112(1) (2002), pp. 83–116. [MV07] H. L. Montgomery and R. C. Vaughan.Multiplicative Number Theory I. Classical Theory. Cambridge ...

work page 1924
[3]

On small zeros of automorphicL-functions

1In view of (39), the bound (40) is equivalent to ζ(σ)> σ2 2σ2 −3σ+ 1 , σ >1. REFERENCES 17 [Oma14] S. Omar. “On small zeros of automorphicL-functions”. In:C. R. Math. Acad. Sci. Paris352 (7-8) (2014), pp. 551–556. [PZ25] N. Paloj¨ arvi and T. Zhao.On Turing’s method for ArtinL-functions and the Selberg class. Preprint,https://arxiv.org/abs/2508.03023

work page arXiv 2014
[4]

On the Phragm´ en-Lindel¨ of theorem and some applications

[Rad59] H. Rademacher. “On the Phragm´ en-Lindel¨ of theorem and some applications”. In:Math. Z.72 (1959/60), pp. 192–204. [Sie45] C. L. Siegel. “On the zeros of the DirichletL-functions”. In:Ann. of Math.46 (3) (1945), pp. 409–422. [Tit86] E. C. Titchmarsh.The Theory of the Riemann Zeta-function, 2nd ed.New York: The Clarendon Press, Oxford University Press,

work page 1959
[5]

Zhao.The positivity technique and low-lying zeros of DirichletL-functions

[Zha25] T. Zhao.The positivity technique and low-lying zeros of DirichletL-functions. Preprint,https://arxiv.org/abs/2503.15832

work page arXiv

[1] [1]

Counting zeros of DirichletL-functions

[Ben+21] M. A. Bennett, G. Martin, K. O’Bryant, and A. Rechnitzer. “Counting zeros of DirichletL-functions”. In:Math. Comp.90(329) (2021), pp. 1455–1482. [Bob+15] J. Bober, J. B. Conrey, D. W. Farmer, A. Fujii, S. Koutsoliotas, S. Lemurell, M. Rubinstein, and H. Yoshida. “The highest lowest zero of generalL-functions”. In:J. Number Theory147 (2015), pp. 3...

work page 2021

[2] [2]

Two notes on the Riemann zeta-function

[Lit24] J. E. Littlewood. “Two notes on the Riemann zeta-function”. In:Proc. Cam- bridge Philos. Soc.22 (1924), pp. 234–242. [Mil02] S. D. Miller. “The highest lowest zero and other applications of positivity”. In: Duke Math. J.112(1) (2002), pp. 83–116. [MV07] H. L. Montgomery and R. C. Vaughan.Multiplicative Number Theory I. Classical Theory. Cambridge ...

work page 1924

[3] [3]

On small zeros of automorphicL-functions

1In view of (39), the bound (40) is equivalent to ζ(σ)> σ2 2σ2 −3σ+ 1 , σ >1. REFERENCES 17 [Oma14] S. Omar. “On small zeros of automorphicL-functions”. In:C. R. Math. Acad. Sci. Paris352 (7-8) (2014), pp. 551–556. [PZ25] N. Paloj¨ arvi and T. Zhao.On Turing’s method for ArtinL-functions and the Selberg class. Preprint,https://arxiv.org/abs/2508.03023

work page arXiv 2014

[4] [4]

On the Phragm´ en-Lindel¨ of theorem and some applications

[Rad59] H. Rademacher. “On the Phragm´ en-Lindel¨ of theorem and some applications”. In:Math. Z.72 (1959/60), pp. 192–204. [Sie45] C. L. Siegel. “On the zeros of the DirichletL-functions”. In:Ann. of Math.46 (3) (1945), pp. 409–422. [Tit86] E. C. Titchmarsh.The Theory of the Riemann Zeta-function, 2nd ed.New York: The Clarendon Press, Oxford University Press,

work page 1959

[5] [5]

Zhao.The positivity technique and low-lying zeros of DirichletL-functions

[Zha25] T. Zhao.The positivity technique and low-lying zeros of DirichletL-functions. Preprint,https://arxiv.org/abs/2503.15832

work page arXiv