Low-Lying Zeros on the Critical Line for Families of Dirichlet $L$-Functions

XinHang Ji

arxiv: 2605.09282 · v2 · pith:CED2MSZNnew · submitted 2026-05-10 · 🧮 math.NT

Low-Lying Zeros on the Critical Line for Families of Dirichlet L-Functions

XinHang Ji This is my paper

Pith reviewed 2026-05-20 23:30 UTC · model grok-4.3

classification 🧮 math.NT MSC 11M26

keywords Dirichlet L-functionslow-lying zeroscritical linemollifierMellin transformshort intervalscharacter sumszero statistics

0 comments

The pith

For large prime moduli P the total number of low-lying zeros on the critical line across all characters mod P exceeds T squared P times square root of log P even when T is as small as 1 over square root of log P.

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

The paper proves a lower bound on the summed count of low-lying zeros lying on the critical line for the family of Dirichlet L-functions modulo a large prime P. For T in the range from a constant over the square root of log P up to 1 the sum over all characters chi mod P of the number N0(T, chi) is at least on the order of T squared P square root of log P. A sympathetic reader cares because earlier methods such as Levinson's or the ordinary Selberg mollifier could not reach these extremely short intervals without losing control of the main term. The work supplies a concrete quantitative statement about zero distribution in this family at scales previously out of reach.

Core claim

We prove that for sufficiently large prime P and real T in the interval [a1 over square root of log P, 1] the sum over chi mod P of N0(T, chi) satisfies sum N0(T, chi) much greater than T squared P square root of log P. The proof proceeds by replacing the usual mollifier with a high-dimensional Mellin transform that disentangles the multi-variable series arising from the mollifier product and thereby extracts a positive main term in the short-interval regime.

What carries the argument

High-dimensional Mellin transform framework that resolves the inseparable cross-terms generated by the mollifier product, allowing extraction of the localized lower bound without error terms that would destroy positivity.

If this is right

The lower bound holds for intervals shorter than those reachable by the Levinson method or by standard Selberg mollifiers.
The same framework supplies a method that can be applied to the zero statistics of higher-rank L-function families.
The short-interval bottleneck in the analysis of low-lying zeros is circumvented for this family of L-functions.

Where Pith is reading between the lines

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

The method may extend to give lower bounds on the proportion of zeros on the critical line for the same family at still smaller scales.
Similar high-dimensional Mellin techniques could be tested on quadratic character families or on families of higher-degree L-functions where cross-term problems also appear.
If the framework generalizes it would supply quantitative evidence for random-matrix predictions about zero spacing at the very bottom of the spectrum.

Load-bearing premise

The high-dimensional Mellin transform resolves the cross-terms from the mollifier calculations without introducing errors large enough to cancel the lower bound in the short-interval regime.

What would settle it

A direct numerical count, for a concrete large prime P and a concrete T near 1 over square root of log P, showing that the summed number of critical-line zeros is smaller than c T squared P square root of log P for any fixed positive c.

read the original abstract

In this paper, we establish a new lower bound for the number of low-lying zeros of Dirichlet $L$-functions $L(s, \chi)$ on the critical line within extremely short intervals. Specifically, for a sufficiently large prime $P$ and real number $T \in [a_1/\sqrt{\log P}, 1]$, we prove that the sum of the number of zeros on the critical line $N_0(T, \chi)$ over characters $\chi \bmod P$ satisfies $$ \sum_{\chi \bmod P} N_0(T, \chi) \gg T^2 P\sqrt{\log P} .$$ Traditional approaches encounter significant technical barriers in this short-interval regime. The Levinson method fails due to its own inherent limitations in handling such restricted intervals , while standard applications of the Selberg mollifier are hindered by the emergence of complex, inseparable cross-terms that are difficult to evaluate. To overcome these obstacles, we introduce a novel analytic framework utilizing high-dimensional Mellin transforms. This approach systematically manages the multi-variable series generated by the mollifier calculations. By explicitly resolving these cross-term obstructions, we extract the localized lower bound, providing a robust method that circumvents the short-interval bottleneck and offers potential applicability to the zero statistics of higher-rank $L$-function families.

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

The paper claims a lower bound on summed low-lying zeros for Dirichlet L-functions in intervals down to 1/sqrt(log P) by using high-dimensional Mellin transforms to separate mollifier cross-terms.

read the letter

The punchline is that this work gets a lower bound on the number of low-lying zeros in intervals shorter than what previous methods reach, specifically down to length 1 over sqrt(log P), by introducing high-dimensional Mellin transforms to manage the cross terms that appear when you square the mollifier. What is new is the systematic use of multi-variable Mellin analysis to resolve those inseparable products in the averaged quadratic form. The paper explains how Levinson's method and standard Selberg mollifiers run into trouble here, and positions the new framework as a way around it. That part is clear and direct. It does well in identifying the technical barrier and sketching a path forward that could extend to other families of L-functions. The soft spots are around the actual implementation of the high-dimensional contours. Shifting contours in several complex variables brings in truncation errors and horizontal integrals, and for T at the bottom of the range, those need to be shown smaller than the main term of size T squared times sqrt(log P). The stress-test raises a fair point that if the uniformity assumptions don't hold tightly enough, the lower bound could fail. Without the explicit estimates in front of me, I can't tell how close it comes, but that's the spot that needs the most attention in a review. This paper is for analytic number theorists who work on zero statistics, mollifiers, and random matrix predictions. A reader who knows the Levinson and Selberg literature would see the value in this adaptation. It deserves a serious referee because the claim is concrete and the method is a fresh angle on a known obstruction. I would recommend putting it through peer review to get feedback on whether the multi-variable error terms are controlled properly.

Referee Report

2 major / 2 minor

Summary. The paper claims to prove that for sufficiently large prime P and T in the interval [a1/sqrt(log P), 1], the sum over characters χ mod P of N0(T, χ) (the number of low-lying zeros on the critical line in an interval of length T) satisfies sum_χ N0(T, χ) ≫ T² P sqrt(log P). The proof introduces a high-dimensional Mellin transform framework to disentangle the cross-terms that arise when applying a Selberg mollifier in this short-interval regime, where Levinson's method and standard one-variable Mellin analysis are said to fail.

Significance. If the central claim holds with the stated error control, the result would constitute a meaningful advance in the distribution of low-lying zeros for families of Dirichlet L-functions. It would be the first explicit lower bound of this strength in intervals as short as 1/sqrt(log P), and the high-dimensional Mellin technique might extend to other L-function families where cross-term obstructions appear.

major comments (2)

The manuscript supplies no explicit comparison of the truncation and horizontal-integral error terms arising from the multi-variable contour shifts against the main-term size T² sqrt(log P) when T is at the lower endpoint a1/sqrt(log P). Without such a comparison, it is impossible to confirm that the averaged quadratic form remains positive after mollification.
The handling of the high-dimensional Mellin integrals is presented at a level of generality that does not include concrete uniformity statements or residue calculations sufficient to bound the contribution of the inseparable cross-terms uniformly down to T ~ 1/sqrt(log P).

minor comments (2)

The abstract would be clearer if it indicated the precise dimension of the Mellin transform or the form of the mollifier employed.
Notation for N0(T, χ) should be defined explicitly in the introduction rather than assumed from prior literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the significance of our results and for the careful reading of the manuscript. The comments highlight important points regarding the clarity of our error estimates in the short-interval regime. We address each major comment below and will incorporate revisions to strengthen the exposition and verifiability of the bounds.

read point-by-point responses

Referee: The manuscript supplies no explicit comparison of the truncation and horizontal-integral error terms arising from the multi-variable contour shifts against the main-term size T² sqrt(log P) when T is at the lower endpoint a1/sqrt(log P). Without such a comparison, it is impossible to confirm that the averaged quadratic form remains positive after mollification.

Authors: We agree that an explicit comparison of these error terms is essential for confirming the positivity of the quadratic form at the lower endpoint. In the revised manuscript, we will add a dedicated subsection (new Section 4.3) that explicitly compares the truncation and horizontal integral contributions to the main term T² P sqrt(log P). Specifically, we bound the combined error by O(T² P sqrt(log P) / log log P) uniformly for T ≥ a1/sqrt(log P), which is o of the main term and preserves the positivity after mollification. The estimates follow from the rapid decay properties of the high-dimensional Mellin transforms and standard convexivity bounds on the relevant Dirichlet series. revision: yes
Referee: The handling of the high-dimensional Mellin integrals is presented at a level of generality that does not include concrete uniformity statements or residue calculations sufficient to bound the contribution of the inseparable cross-terms uniformly down to T ~ 1/sqrt(log P).

Authors: We acknowledge that the current presentation of the high-dimensional Mellin integrals would benefit from more concrete uniformity statements and explicit residue calculations to control the cross-terms down to the lower endpoint. In the revision, we will expand Theorem 2.1 to include explicit uniformity in all parameters (including the height T and the dimension of the Mellin transform), and we will add an appendix with detailed residue computations for the inseparable cross-terms. These calculations show that the cross-term contributions are bounded by O(T^{2-δ} P sqrt(log P)) for a small positive δ depending only on the mollifier length, uniformly in the stated range of T. This ensures they remain smaller than the main term. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new framework

full rationale

The paper derives the lower bound sum_χ N0(T, χ) ≫ T² P √(log P) for T in [a1/√(log P), 1] by introducing a high-dimensional Mellin transform framework that resolves inseparable cross-terms from the mollifier square. This is presented as an independent analytic innovation that overcomes limitations of Levinson and standard Selberg mollifiers, without any quoted reduction of the main term or positivity to a fitted parameter, self-citation chain, or definitional tautology. No load-bearing step equates the claimed result to its inputs by construction, and the approach is self-contained against external benchmarks as a direct consequence of the new multi-variable contour analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard background facts of analytic number theory (functional equation, Euler product, approximate functional equation for Dirichlet L-functions) and introduces a new technical framework; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

standard math Dirichlet L-functions satisfy the standard functional equation and have an Euler product.
These are the usual analytic properties invoked for all zero-counting arguments in this area.

pith-pipeline@v0.9.0 · 5763 in / 1305 out tokens · 34821 ms · 2026-05-20T23:30:23.341168+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 3.1 and the subsequent definition of the 4-fold Mellin transform M(s1,s2,s3,s4) together with the quadruple Dirichlet series F(s1,s2,s3,s4) used to disentangle the cross-term in S(θ).
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Estimation of S(θ) via contour integration and the resulting bound S(θ) ≪ X^{2θ}/√log X that feeds the L² bounds on Iχ and Jχ.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages · 1 internal anchor

[1]

Selberg, On the zeros of Riemann’s zeta-function,Skr

A. Selberg, On the zeros of Riemann’s zeta-function,Skr. Norske Vid. Akad. Oslo I, no. 10 (1942), 1–59

work page 1942
[2]

E. C. Titchmarsh,The Theory of the Riemann Zeta-function, second edition, revised by D. R. Heath-Brown, Clarendon Press, Oxford, 1986

work page 1986
[3]

Dickinson, Zeros of dirichlet L-functions near the critical line,Mathematika70 (2024):e12239

G. Dickinson, Zeros of dirichlet L-functions near the critical line,Mathematika70 (2024):e12239 . https://doi.org/10.1112/mtk.12239 https://arxiv.org/abs/2211.06264

work page doi:10.1112/mtk.12239 2024
[4]

Levinson, More than one-third of the zeros of Riemann’s zeta-function are onσ= 1 2,Adv

N. Levinson, More than one-third of the zeros of Riemann’s zeta-function are onσ= 1 2,Adv. Math.13 (1974), 383–436. https://doi.org/10.1016/0001-8708(74)90074-7

work page doi:10.1016/0001-8708(74)90074-7 1974
[5]

J. B. Conrey, More than two fifths of the zeros of the Riemann zeta function are on the critical line,J. Reine Angew. Math., 399 (1989), 1–26. https://doi.org/10.1515/crll.1989.399.1

work page doi:10.1515/crll.1989.399.1 1989
[6]

J. B. Conrey, Henryk Iwaniec, K. Soundrarajan, Critical zeros of Dirichlet L-functions, J. reine angew. Math., 681 (2013), 175–198. https://doi.org/10.1515/crelle-2012-0032 https://arxiv.org/abs/1105.1177 28

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1515/crelle-2012-0032 2013

[1] [1]

Selberg, On the zeros of Riemann’s zeta-function,Skr

A. Selberg, On the zeros of Riemann’s zeta-function,Skr. Norske Vid. Akad. Oslo I, no. 10 (1942), 1–59

work page 1942

[2] [2]

E. C. Titchmarsh,The Theory of the Riemann Zeta-function, second edition, revised by D. R. Heath-Brown, Clarendon Press, Oxford, 1986

work page 1986

[3] [3]

Dickinson, Zeros of dirichlet L-functions near the critical line,Mathematika70 (2024):e12239

G. Dickinson, Zeros of dirichlet L-functions near the critical line,Mathematika70 (2024):e12239 . https://doi.org/10.1112/mtk.12239 https://arxiv.org/abs/2211.06264

work page doi:10.1112/mtk.12239 2024

[4] [4]

Levinson, More than one-third of the zeros of Riemann’s zeta-function are onσ= 1 2,Adv

N. Levinson, More than one-third of the zeros of Riemann’s zeta-function are onσ= 1 2,Adv. Math.13 (1974), 383–436. https://doi.org/10.1016/0001-8708(74)90074-7

work page doi:10.1016/0001-8708(74)90074-7 1974

[5] [5]

J. B. Conrey, More than two fifths of the zeros of the Riemann zeta function are on the critical line,J. Reine Angew. Math., 399 (1989), 1–26. https://doi.org/10.1515/crll.1989.399.1

work page doi:10.1515/crll.1989.399.1 1989

[6] [6]

J. B. Conrey, Henryk Iwaniec, K. Soundrarajan, Critical zeros of Dirichlet L-functions, J. reine angew. Math., 681 (2013), 175–198. https://doi.org/10.1515/crelle-2012-0032 https://arxiv.org/abs/1105.1177 28

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1515/crelle-2012-0032 2013