arxiv: 2605.09282 · v1 · submitted 2026-05-10 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

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

XinHang Ji

Pith reviewed 2026-05-12 03:54 UTC · model grok-4.3

classification 🧮 math.NT

keywords Dirichlet L-functionslow-lying zeroscritical lineSelberg mollifierMellin transformsshort intervalszero distributionprime modulus

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The pith

For large prime P, the sum over characters mod P of low-lying zeros of L(s, chi) on the critical line in intervals of length T is at least order T squared times P times sqrt(log P), even for T as small as 1 over sqrt(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 total number of low-lying zeros lying on the critical line for the family of all Dirichlet L-functions with characters modulo a large prime P. It shows that when the interval length T is at least a constant over the square root of log P, the summed count N0(T, chi) across all such characters is asymptotically larger than T squared P times the square root of log P. Traditional tools such as Levinson's method or the ordinary Selberg mollifier cannot reach this short-interval range because of technical obstructions from cross-terms. The author therefore develops a high-dimensional Mellin transform that organizes the multi-variable series produced by the mollifier and extracts the desired lower bound.

Core claim

For a sufficiently large prime P and real number T in the interval [a1/sqrt(log P), 1], the sum over all characters chi modulo P of N0(T, chi) satisfies sum N0(T, chi) ≫ T² P sqrt(log P). This lower bound is obtained by introducing a high-dimensional Mellin transform framework that systematically resolves the multi-variable series arising from the mollifier calculations and thereby removes the cross-term obstructions that block earlier approaches.

What carries the argument

A high-dimensional Mellin transform framework that organizes and evaluates the multi-variable series generated by the Selberg mollifier to isolate the main-term contribution without residual cross-term errors.

If this is right

The bound holds in an interval-length regime where both Levinson's method and standard applications of the Selberg mollifier are blocked by cross-term difficulties.
The same framework supplies a route to zero-distribution statistics at scales shorter than those previously accessible for this family.
The technique is presented as potentially applicable to the zero statistics of families of higher-rank L-functions.

Where Pith is reading between the lines

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

The same transform method might be applied to obtain lower bounds on zeros for other families of L-functions whose mollifiers produce analogous multi-variable series.
If the bound is sharp, it would suggest that a positive proportion of the characters modulo P possess at least one zero in each such short interval.
Explicit numerical checks on moderate-sized primes could be used to test the range of validity of the constant a1 and the implied constant in the lower bound.

Load-bearing premise

The high-dimensional Mellin transform fully controls the multi-variable series from the mollifier so that no leftover cross-terms or error terms invalidate the lower-bound extraction when T is as small as 1 over sqrt(log P).

What would settle it

Direct numerical computation, for a concrete large prime P such as 10^7, of the actual sum of N0(T, chi) for T equal to 1/sqrt(log P) would falsify the claim if the computed sum lies below any positive multiple of T² P sqrt(log P).

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 new short-interval lower bound for summed low-lying zeros of Dirichlet L-functions using high-dimensional Mellin transforms, but error control details are crucial to check.

read the letter

The main thing to know is that this paper claims a lower bound sum over chi mod P of N0(T, chi) much larger than T squared P sqrt(log P) for T down to about 1 over sqrt(log P). They reach this by applying high-dimensional Mellin transforms to the multi-variable series from the Selberg mollifier, which is meant to untangle the cross-terms that block standard approaches in such short ranges. Levinson's method hits its own limits here, and ordinary mollifiers produce inseparable terms that are hard to evaluate when the interval shrinks that far. The new framework is presented as a direct way to manage those terms and pull out the localized count. What stands out as new is the systematic high-dimensional Mellin device for handling the cross-term obstructions in the Dirichlet family at this scale. The abstract does a clear job laying out the barriers and sketching how the transform resolves the multi-variable expressions to extract the bound. It does well at identifying the exact technical bottleneck and suggesting the method could carry over to other L-function families. The approach looks like a genuine addition to the toolkit rather than a minor tweak. The soft spot is the missing detail on error control. At the lower end of the T range, any unresolved cross-term or truncation error from the contour shifts that is not o of the main term would invalidate the lower bound. The abstract asserts that the transforms explicitly resolve the obstructions, but without explicit estimates or verification that new errors from the high-dimensional residues stay small enough, the claim rests on an unshown step. This matches the stress-test concern exactly. Readers who follow analytic number theory work on zero distributions, mollifiers, and random matrix models would get the most value here. It is aimed at specialists who can assess whether the transform application actually delivers uniform bounds in the stated regime. I would send it for peer review. The result is concrete, the method is new enough to matter, and the area is active, so referees should check the details on the error terms and contour choices.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to prove that for sufficiently large primes P and T in the interval [a1/sqrt(log P), 1], the sum over characters chi mod P of N0(T, chi) satisfies sum N0(T, chi) >> T^2 P sqrt(log P). This is achieved by applying a novel high-dimensional Mellin transform framework to the multi-variable series arising from a Selberg mollifier, which is asserted to explicitly resolve the cross-term obstructions that block standard mollifier arguments in this short-interval regime.

Significance. If the central lower bound holds with the claimed uniformity, the result would be a notable advance in the distribution of low-lying zeros for Dirichlet L-function families, pushing the known range for positive lower bounds on the number of zeros in short intervals well below the thresholds accessible by Levinson's method or classical mollifiers. The introduction of high-dimensional Mellin transforms as a tool for managing inseparable cross-terms in mollified sums could have wider applicability to zero statistics for higher-rank L-functions. The paper supplies a direct analytic construction rather than relying on fitted constants.

major comments (2)

[Abstract / main derivation] Abstract and proof outline: the assertion that the high-dimensional Mellin transform 'explicitly resolv[es] these cross-term obstructions' is load-bearing for the lower bound, yet no explicit error estimates, contour choices, or residue computations are supplied to confirm that all remainder terms (including truncation and cross-term contributions) are o(T^2 P sqrt(log P)) uniformly down to T = a1/sqrt(log P). Without these, the extraction of a positive main term cannot be verified.
[Mollifier calculation and transform application] Application to the mollifier series: the multi-variable series generated by the Selberg mollifier produces overlapping support in the short-interval regime; the manuscript must demonstrate that the high-dimensional contour shifts cancel or bound all cross-terms without introducing new errors of size comparable to the main term. The current description leaves this step unverified.

minor comments (1)

[Abstract] The abstract would benefit from a brief indication of the dimension of the Mellin transform and the precise form of the weight functions in the mollifier to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and valuable comments on our manuscript. We address each major comment below and are prepared to revise the paper to strengthen the presentation of the technical details.

read point-by-point responses

Referee: [Abstract / main derivation] Abstract and proof outline: the assertion that the high-dimensional Mellin transform 'explicitly resolv[es] these cross-term obstructions' is load-bearing for the lower bound, yet no explicit error estimates, contour choices, or residue computations are supplied to confirm that all remainder terms (including truncation and cross-term contributions) are o(T^2 P sqrt(log P)) uniformly down to T = a1/sqrt(log P). Without these, the extraction of a positive main term cannot be verified.

Authors: The referee correctly identifies that the verification of the error terms is central to the argument. In the manuscript the high-dimensional Mellin transform is constructed so that the contours can be shifted independently in each variable to regions of absolute convergence, with the cross-term contributions controlled by the resulting exponential decay. The main term is extracted from the residues at the relevant poles, and standard bounds on the Dirichlet series and Gamma factors are used to show the remainders are smaller than the main term by a factor of (log P)^{-c} for some c>0, uniformly for T down to a1/sqrt(log P). To make this verification immediate for the reader, we will add an explicit subsection detailing the contour choices, the residue computations, and the uniform error estimates in the revised version. revision: yes
Referee: [Mollifier calculation and transform application] Application to the mollifier series: the multi-variable series generated by the Selberg mollifier produces overlapping support in the short-interval regime; the manuscript must demonstrate that the high-dimensional contour shifts cancel or bound all cross-terms without introducing new errors of size comparable to the main term. The current description leaves this step unverified.

Authors: The Selberg mollifier indeed generates a multi-variable Dirichlet series whose support overlaps when T is as small as 1/sqrt(log P). Our framework applies the high-dimensional Mellin transform to factor the series into a product of one-variable transforms, after which each contour is shifted separately. The off-diagonal (cross-term) contributions are then bounded by moving the contours far to the left, where the rapid decay of the Mellin transform dominates the growth of the L-functions and the mollifier coefficients. The resulting error is absorbed into the o(T^2 P sqrt(log P)) term. We agree that the current exposition would benefit from a more expanded calculation of these bounds. In the revision we will insert a detailed verification of the contour shifts and the size of the cross-term integrals, confirming that no new errors of main-term size appear. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on novel analytic construction without reduction to inputs

full rationale

The paper derives the lower bound sum_{chi mod P} N_0(T, chi) >> T^2 P sqrt(log P) for T down to a1/sqrt(log P) by introducing a high-dimensional Mellin transform framework applied to the multi-variable series arising from the Selberg mollifier. This framework is presented as a direct analytic tool that resolves cross-term obstructions via contour integration and residue computations, without any quoted steps that define the target quantity in terms of itself, rename fitted parameters as predictions, or invoke load-bearing self-citations whose validity depends on the present result. The abstract and described method establish an independent construction that extracts the localized lower bound from the transformed series, remaining self-contained against external benchmarks such as standard mollifier limitations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no free parameters or new entities are mentioned. The work rests on standard analytic number theory background for Dirichlet L-functions and the functional equation.

axioms (1)

domain assumption Dirichlet L-functions satisfy the standard functional equation and Euler product representation used to define N0(T, chi) and to apply mollifiers.
Implicit in the definition of the zero-counting function and the mollifier calculations referenced in the abstract.

pith-pipeline@v0.9.0 · 5532 in / 1511 out tokens · 74569 ms · 2026-05-12T03:54:31.609772+00:00 · methodology

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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
high-dimensional Mellin transforms... explicitly resolving these cross-term obstructions... S(θ) = O(X^{2θ}/√log X)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery unclear
Theorem 1.1... sum N_0(T,χ) ≫ T² P √log P for T ≥ a1/√log P

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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