pith. sign in

arxiv: 2512.14907 · v2 · submitted 2025-12-16 · 🧮 math.NT

Unconditional estimates on the argument of Dirichlet L-functions with applications to low-lying zeros

Ghaith Hiary , Tianyu Zhao This is my paper

Pith reviewed 2026-05-16 21:25 UTC · model grok-4.3

classification 🧮 math.NT MSC 11M0611M26
keywords Dirichlet L-functionslow-lying zerosargument estimatesSelberg theoremexplicit boundsprime moduluszero distribution
0
0 comments X

The pith

For all large primes q, the first non-trivial zero of Dirichlet L-functions mod q lies below 1075 times the average spacing.

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

The paper makes an explicit version of Selberg's averaged estimate on the argument of Dirichlet L-functions over characters modulo a prime q. This explicit bound is then used to show that in the family of L-functions for such q, the lowest non-trivial zero has imaginary part less than 1075 times 2π over log q, the typical spacing of low-lying zeros. A lower bound is also given on how many of these L-functions have their first zero within a fixed multiple of that spacing. These results are unconditional and fully explicit, providing concrete numbers where previous work had only existence or conditional statements.

Core claim

By deriving explicit constants in Selberg's result on the average of the argument function of L(s, χ) for non-principal characters χ modulo prime q, the authors prove that for all sufficiently large such q the smallest height of a non-trivial zero in the family is at most 1075 · (2π / log q). They further establish a positive lower bound on the proportion of characters for which the first zero lies within a specified multiple of the average spacing.

What carries the argument

The explicit averaged estimate on the argument of L(s, χ) over characters mod q, which controls the distribution of zeros near the central point.

Load-bearing premise

The explicit form of Selberg's averaged argument result holds with the paper's derived constants once q is a sufficiently large prime.

What would settle it

Finding a prime q large enough that some non-principal character mod q has its first non-trivial zero at height greater than 1075 · 2π / log q would disprove the claim.

read the original abstract

We make explicit a result of Selberg on the argument of Dirichlet $L$-functions averaged over non-principal characters modulo a prime $q$. As a corollary, we show for all sufficiently large prime $q$ that the height of the lowest non-trivial zero of the corresponding family of $L$-functions is less than $1075\cdot \frac{2\pi}{\log q}$. Here the scaling factor $\frac{2\pi}{\log q}$ is the average spacing between consecutive low-lying zeros with height at most 1, say. We also obtain a lower bound on the proportion of $L$-functions whose first zero lies within a given multiple of the average spacing. These appear to be the first explicit unconditional results of their kinds.

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.

Referee Report

2 major / 2 minor

Summary. The paper makes explicit Selberg's averaged argument result for Dirichlet L-functions over non-principal characters modulo a prime q. As a corollary, it proves that for all sufficiently large primes q the lowest non-trivial zero lies below height 1075 times the mean spacing 2π/log q, and it also gives an explicit lower bound on the proportion of L-functions in the family whose first zero lies within a given multiple of this spacing. These are claimed to be the first such explicit unconditional results.

Significance. If the explicit constants and error-term controls are correct, the work supplies the first unconditional, fully explicit bound on the height of the lowest zero in the prime-modulus Dirichlet family, together with a quantitative proportion statement. The explicit constant 1075 and the direct derivation from Selberg's formula constitute a concrete advance that can be used in further unconditional applications to low-lying zeros.

major comments (2)
  1. [§3] §3, the explicit version of Selberg's averaged argument (leading to the constant 1075): the final numerical value 1075 is obtained after a long chain of explicit estimates; the manuscript must supply a self-contained verification (or computer-assisted check) that no intermediate constant was inadvertently relaxed, since this single number is load-bearing for both the main corollary and the proportion bound.
  2. [Theorem 1.1] Theorem 1.1 and the definition of 'sufficiently large': the statement that the bound holds for all primes q > Q0 requires an explicit (even if large) value of Q0 together with a uniform control of all error terms for q > Q0; without this, the claim that the result is unconditional for 'all sufficiently large prime q' remains formally incomplete.
minor comments (2)
  1. [§2] The notation for the averaged argument function S(χ, t) should be compared explicitly with Selberg's original definition to avoid any ambiguity in the transition from the non-explicit to the explicit setting.
  2. [Introduction] Figure 1 (if present) or the numerical illustration of the proportion bound would benefit from a caption stating the precise range of q used in the plot.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestions. The comments help strengthen the explicitness of our results. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3, the explicit version of Selberg's averaged argument (leading to the constant 1075): the final numerical value 1075 is obtained after a long chain of explicit estimates; the manuscript must supply a self-contained verification (or computer-assisted check) that no intermediate constant was relaxed, since this single number is load-bearing for both the main corollary and the proportion bound.

    Authors: We agree that a transparent, self-contained verification of the constant 1075 is essential. In the revised manuscript we will add a new appendix that reproduces the full chain of explicit estimates, listing every intermediate constant and the inequalities used to obtain it. This appendix will allow direct verification that no relaxation occurred. If any step relies on a short computer-assisted check, we will state the precise computation performed and the software used. revision: yes

  2. Referee: [Theorem 1.1] Theorem 1.1 and the definition of 'sufficiently large': the statement that the bound holds for all primes q > Q0 requires an explicit (even if large) value of Q0 together with a uniform control of all error terms for q > Q0; without this, the claim that the result is unconditional for 'all sufficiently large prime q' remains formally incomplete.

    Authors: We accept the point. In the revised version we will compute and state an explicit numerical value for Q0 (derived from the error-term bounds already present in the paper) such that the stated inequalities hold uniformly for all primes q > Q0. The proof of Theorem 1.1 will be updated to include this explicit threshold and the verification that all error terms are controlled beyond it. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit version of external Selberg result

full rationale

The paper derives explicit constants for Selberg's averaged argument theorem on Dirichlet L-functions modulo prime q, then applies the result to bound the lowest non-trivial zero height below 1075 mean spacings for large q. This is a direct, parameter-free explicitization of an external classical theorem (Selberg, not self-citation), with error terms controlled in the large-q regime. No step reduces by construction to a fitted input, self-definition, or author-specific uniqueness theorem; the central bound follows from the explicit estimates without renaming known patterns or smuggling ansatzes. The derivation remains self-contained against the independent Selberg benchmark.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard analytic number theory tools for L-functions (functional equation, approximate functional equation, character sum estimates) plus the non-explicit Selberg result that is made explicit here. The constant 1075 is derived rather than fitted to data.

free parameters (1)
  • 1075
    Explicit numerical constant obtained by bounding all error terms when making Selberg's averaged argument result effective; appears in the final zero-height bound.
axioms (1)
  • domain assumption Selberg's theorem on the average argument of Dirichlet L-functions over characters mod q
    The paper starts from this known result and renders it explicit with concrete constants.

pith-pipeline@v0.9.0 · 5424 in / 1231 out tokens · 49090 ms · 2026-05-16T21:25:08.938226+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

  1. [1]

    Zeros of DirichletL-functions

    [BM92] R. Balasubramanian and V. K. Murty. “Zeros of DirichletL-functions”. In:Ann. Sci. ´Ecole Norm. Sup. (4)25(5) (1992), pp. 567–615. [Bui12] H. M. Bui. “Non-vanishing of DirichletL-functions at the central point”. In:Int. J. Number Theory8(8) (2012), pp. 1855–1881. [CCM22] E. Carneiro, A. Chirre, and M. B. Milinovich. “Hilbert spaces and low-lying zer...

    work page 1992

  2. [2]

    One-level density estimates for Dirich- let L-functions with extended support

    [DPR23] S. Drappeau, K. Pratt, and M. Radziwi l l. “One-level density estimates for Dirich- let L-functions with extended support”. In:Algebra Number Theory17(4) (2023), pp. 805–830. 28 REFERENCES [FM13] D. Fiorilli and G. Martin. “Inequities in the Shanks–R´ enyi prime number race: An asymptotic formula for the densities”. In:J. reine angew. Math.676 (20...

    work page 2023

  3. [3]

    Explicit zero-free regions for the Riemann zeta-function

    [MTY24] M. J. Mossinghoff, T. S. Trudgian, and A. Yang. “Explicit zero-free regions for the Riemann zeta-function”. In:Res. Number Theory10(1), Paper No. 11, 27pp (2024). [Mur90] M. R. Murty. “On simple zeros of certainL-series”. In:Number Theory (Banff, AB, 1988). de Gruyter, Berlin, 1990, pp. 427–439. [OS99] A. E. ¨Ozl¨ uk and C. Snyder. “On the distrib...

    work page 2024

  4. [4]

    Average non-vanishing of DirichletL-functions at the central point

    [Pra19] K. Pratt. “Average non-vanishing of DirichletL-functions at the central point”. In:Algebra Number Theory13(1) (2019), pp. 227–249. [QW25] X. Qin and X. Wu.Non-vanishing of DirichletL-functions at the Central Point. Preprint,https://arxiv.org/abs/2504.11916

    work page arXiv 2019

  5. [5]

    Contributions to the theory of Dirichlet’sL-functions

    [Sel46] A. Selberg. “Contributions to the theory of Dirichlet’sL-functions”. In:Skr. Norske Vid.-Akad. Oslo I3 (1946). [Sit85] R. Sitaramachandra. “On an error term of Landau. II.” In:Rocky Mountain J. Math.15(2) (1985), pp. 579–588. [Sou00] K. Soundararajan. “Nonvanishing of quadratic DirichletL-functions ats= 1 2”. In:Ann. of Math. (2)152(2) (2000), pp....

    work page 1946

  6. [6]

    Zhao.Conditional estimates on the argument of Dirichlet L-functions with applications to low-lying zeros

    [Zha25a] T. Zhao.Conditional estimates on the argument of Dirichlet L-functions with applications to low-lying zeros. Preprint,https://arxiv.org/abs/2508.13301

    work page arXiv

  7. [7]

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

    [Zha25b] T. Zhao.The positivity technique and low-lying zeros of DirichletL-functions. Preprint,https://arxiv.org/abs/2503.15832

    work page arXiv