Short mollifiers of the Riemann zeta-function

Caroline L. Turnage-Butterbaugh; Chung-Hang Kwan; David W. Farmer; J. Brian Conrey; Yongxiao Lin

REVIEW 3 major objections 1 minor 1 cited by

Short mollifiers of the Riemann zeta-function

T0 review · 3 major / 1 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper claims that a calculus-of-variations choice of linear combinations of derivatives of the Riemann zeta function yields a positive proportion of zeros on the critical line even when the mollifier is arbitrarily short, and that the s

desk verdict The abstract promises a major Levinson-method result, but the full text is a completely different computer science paper, so nothing here is auditable. read the letter →

arxiv 2508.11108 v1 pith:EYK5TL42 submitted 2025-08-14 math.NT

J. Brian Conrey , David W. Farmer , Chung-Hang Kwan , Yongxiao Lin , Caroline L. Turnage-Butterbaugh This is my paper

classification math.NT MSC 11M0611M2611M41

keywords RiemannzetafunctioncriticallineLevinson'smethodmollifiercalculusofvariationsmodularL-functionsSiegel'sf-functionzeros

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Levinson's method proves that a positive proportion of the nontrivial zeros of the Riemann zeta function lie on the critical line by mollifying the function with a short Dirichlet polynomial; classically, the proved proportion shrinks as the mollifier gets shorter. This paper shows that if one replaces the usual mollifier by an optimized linear combination of derivatives of zeta—chosen through the calculus of variations—the proportion need not go to zero, no matter how short the mollifier is. The same construction applies to modular L-functions, and with Levinson's original mollifier it more than doubles the previously known zero proportions for those functions. The paper attributes the gain to optimizing the linear combination rather than to refining the mollifier, and notes a surprising connection to Siegel's $\mathfrak{f}$-function in the Riemann–Siegel formula.

What carries the argument

The central object is a sequence of linear combinations of the derivatives $\zeta^{(k)}(s)$ (and similarly for the L-functions), with coefficients determined by a calculus-of-variations optimization. This combination plays the role of Levinson's mollifier; its behavior under the relevant moments is what Levinson's method needs. The variational principle selects coefficients that make the mollified function's mean square and related averages have the right positivity, and it is this optimization that sustains a positive zero proportion even when the mollifier is arbitrarily short.

What would settle it

Work out the variational equations at zero mollifier length; if the predicted positive limiting proportion is not actually achieved when plugging the known moment main terms into Levinson's formula, or if a numerical check of a specific modular L-function gives a proportion at or below the previous bound, the claim would be falsified. Concretely: compute the paper's proportion for the first few weight and level families for which the arithmetic inputs are known and compare with the doubled bound.

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Extended reading notes

Core claim

The central claim is that there exists a sequence of short mollifiers, built from linear combinations of derivatives of $\zeta$, for which Levinson's method recovers a positive proportion of critical-line zeros uniformly as the mollifier length tends to zero. The coefficients of the linear combination are chosen as the solution of a variational problem, and the paper argues that this optimization, not the shape of the mollifier, is what keeps the proportion bounded away from zero. For modular $L$-functions, the same construction gives proportions that more than double the earlier results of Bernard and Kühn–Robles–Zeindler while using the same arithmetic moment inputs. The paper also observe

Load-bearing premise

The conclusion depends on the moment asymptotics for the zeta function and for the modular L-functions holding at very short mollifier lengths with the precision that the variational calculation requires; if any of those main terms is wrong or merely conjectural for the families used, the derived proportions would not follow.

Editorial extensions

If this is right

If correct, Levinson-type methods no longer suffer from the usual 'short mollifier' limitation: a positive share of critical-line zeros can be established without needing a long mollifier.
The same construction applies to modular L-functions, doubling previously known critical-line zero proportions while relying only on the established arithmetic input (moment asymptotics).
The result redirects attention from mollifier design to optimizing the linear combinations of derivatives, a comparatively neglected component of Levinson's method.
The connection to Siegel's $\mathfrak{f}$-function provides a new analytic handle on the Riemann–Siegel formula, potentially linking the variational construction to classical approximations of $\zeta$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural test is whether the same variational optimization can extend to other automorphic L-functions once analogues of the required moment asymptotics are known; if the only input is those moments, the method might generalize broadly.
The link to Siegel's $\mathfrak{f}$-function suggests that the optimized combinations might be interpreted as approximate functional equations, which could be used to build numerical test functions for locating zeros.
If the positive-proportion claim is robust, it may imply that the obstacle to extending Levinson's method to longer ranges is not the mollifier length but the accuracy of the moment expansions; the variational viewpoint gives a way to measure how much precision is needed.
Because the stated proportions for modular L-functions depend on the same arithmetic inputs as previous work, an independent verification of the doubling would settle the method's merit without relying on the variational details.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know one thing first: the manuscript body is not the paper in the abstract. arXiv 2508.11108 is listed as a math.NT paper by Conrey, Farmer, Kwan, Lin, and Turnage-Butterbaugh, and the abstract is a clear, strong claim about short mollifiers and positive proportions of critical-line zeros. But the full text is a NeurIPS submission by Singh et al. on code diffusion models. That is not a missing appendix or a typo; it is a different paper, with different authors, different notation, and different subject. That alone kills any chance of refereeing this version.

What the abstract actually proposes is interesting. The idea of optimizing the linear combination of derivatives inside Levinson's method, rather than the mollifier length, is a sensible conceptual shift. The reported more-than-doubling of known proportions for modular L-functions, if correct, would be a real quantitative improvement, and the connection to Siegel's f-function is the kind of byproduct that makes a paper worth reading. The authors are not obviously wrong from the abstract alone.

But the soft spots are not small. The stress-test note is on target: the claim "regardless of how short the mollifier is" needs a uniform lower bound on the Levinson functional as the mollifier length tends to zero, and the abstract gives no equation, no Euler-Lagrange statement, no error term, nothing to check. Standard short-mollifier choices make that functional tend to zero, so the whole result hangs on an optimizer that stays positive and yields a bounded-below integral. That is a plausible mechanism, but it is unverifiable here. The modular L-function doubling also inherits the same arithmetic moment assumptions, which are usually only established for sufficiently long mollifiers; extending them to the arbitrarily short regime is exactly where the risk lies.

Still, the dominant problem is the text mismatch. You cannot credit the math because the math is not present. You cannot even tell whether the abstract is a ghost or the body is a paste error. A serious editor would desk-reject this submission and ask for the correct full text. If the correct manuscript shows up and matches the abstract, the work may well deserve careful refereeing. But this version does not.

Referee Report

3 major / 1 minor

Summary. The submission consists of an abstract announcing a calculus-of-variations construction of linear combinations of derivatives of the Riemann zeta-function, adapted to Levinson's method, with the claimed conclusion that a positive proportion of zeros lies on the critical line 'regardless of how short the mollifier is.' It further claims the construction extends to modular L-functions, more than doubling previously known proportions while using the same arithmetic inputs as Bernard and Kühn--Robles--Zeindler, and that the resulting linear combinations provide nontrivial smooth approximations of Siegel's f-function. The body of the received manuscript, however, is an unrelated paper on code-diffusion models for last-mile program repair. No equations, definitions, theorem statements, moment asymptotics, or proofs pertaining to the abstract are present.

Significance. If the abstract's claims were substantiated, the result would be substantial: it would overcome a known difficulty in Levinson's method by showing that optimizing the linear combination, rather than lengthening the mollifier, can preserve a positive proportion of critical-line zeros, and it would give a concrete quantitative improvement for modular L-functions. However, the submission as received contains none of the mathematical content needed to evaluate these claims. It is effectively an abstract-only submission. The uniform 'regardless of how short' claim is historically delicate, and the modular L-function claim depends on arithmetic moment inputs whose validity at short mollifier lengths must be checked. No evidence is supplied for any of these points.

major comments (3)

[Abstract / Full text] The received manuscript contains no mathematical development. There is no definition of the variational problem, no functional being optimized, no Euler--Lagrange equation, no choice of linear combination, no statement of the moment asymptotics used, and no theorem with an error term. The body of the paper is an unrelated document on code-diffusion repair (Sections 1--4 discuss diffusion models and program repair, not zeta-functions). It is impossible to audit the central claim from this material. A journal submission must contain the actual derivation or a precise pointer to it.
[Abstract, first sentence] The quantifier 'regardless of how short the mollifier is' is load-bearing. In Levinson's framework, standard short mollifiers make the recovered proportion tend to zero because the controlling functional degenerates. To support the claimed uniformity, the paper must show that the optimized linear combination has a nondegenerate, uniformly positive weight as the mollifier length tends to zero, and that the relevant moment expansions are valid with errors uniform in that regime. None of these steps appears in the received text.
[Abstract, modular L-function claim] The claim that the construction 'more than doubles' the proportions for modular L-functions while 'relying on the same arithmetic inputs' as Bernard and Kühn--Robles--Zeindler requires specification of the families, the relevant moments, and whether the needed asymptotics are proven or conjectural at the short-mollifier lengths used. If a moment main term is incorrect or only conjectural for the family in question, the derived proportion does not follow. No such specification is present.

minor comments (1)

[Abstract, final sentence] The connection to Siegel's f-function is announced but not developed. If this is a substantive observation, a precise statement and explanation are needed; otherwise it should be removed or marked as a remark.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity demonstrated; abstract reports a variational construction over known arithmetic inputs, with no prediction stated as an input.

full rationale

The only in-scope text for arXiv:2508.11108 is the abstract; the attached full-text PDF is a different paper (arXiv:2508.11110 on code diffusion) and provides no equations for this work. The abstract claims that a new sequence of linear combinations of derivatives of the Riemann zeta-function, adapted to Levinson's method, yields a positive proportion of critical-line zeros 'regardless of how short the mollifier is.' It also claims the construction extends to modular L-functions and 'more than doubles' previously obtained proportions while 'relying on the same arithmetic inputs' as Bernard and Kühn–Robles–Zeindler. None of these statements exhibit a circular reduction: the output proportions are presented as consequences of a variational optimization, not as inputs to that optimization. The phrase 'same arithmetic inputs' is a dependence on prior moment asymptotics, not a self-citation chain. There is no equation in the abstract that would allow one to show the optimized functional is defined in terms of the target proportion, nor any fitted parameter renamed as a prediction. The legitimate concern—that the uniformity as mollifier length tends to zero and the accuracy of the arithmetic inputs are unverified—is a correctness/auditability risk, not a demonstrated circularity. Absent equations, no specific circular step can be quoted, so the score is kept at the low end of the non-circular range.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The abstract exposes few explicit assumptions. The two load-bearing implicit premises are (1) the Levinson framework relating critical-line zero counts to moment functionals, and (2) the validity of the underlying moment asymptotics for ζ and for modular L-functions at very short mollifier lengths ('the same arithmetic inputs' as prior work). No free parameters or invented entities are visible. This ledger is necessarily incomplete: the variational construction itself and its strict-positivity arguments live in the missing body.

assumptions (3)

domain assumption Levinson's method: the proportion of critical-line zeros is governed by the sign and positivity of moment-type integrals of a mollified combination of ζ and its derivatives.
The construction is 'adapted to Levinson's method' (abstract). The method's counting criterion is substantial background and is assumed without proof or recap in the abstract.
domain assumption The relevant moment asymptotics for ζ and for modular L-functions hold to the order needed at very short mollifier lengths ('the same arithmetic inputs' as prior work).
The positive-proportion conclusion is only as strong as these averages; the abstract does not name which arithmetic inputs are used, and for L-functions some may be unproved.
standard math Background facts surrounding the Riemann-Siegel formula used for the final remark about smooth approximations of Siegel's f-function.
The abstract invokes the well-known Riemann-Siegel formula as background; no statement of which facts are used is given.

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Cite this review

Pith. "Pith review of Short mollifiers of the Riemann zeta-function." pith.science (2026). https://pith.science/paper/EYK5TL42

@misc{pith2026250811108,
  author       = {Pith},
  title        = {Pith review of: Short mollifiers of the Riemann zeta-function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EYK5TL42}},
  note         = {Machine review of arXiv:2508.11108}
}

abstract

We apply the calculus of variations to construct a new sequence of linear combinations of derivatives of the Riemann $\zeta$-function adapted to Levinson's method, which yield a positive proportion of zeros of the $\zeta$-function on the critical line, regardless of how short the mollifier is. Our construction extends readily to modular $L$-functions. Even with Levinson's original choice of mollifier, our method more than doubles the proportions of zeros on the critical line for modular $L$-functions previously obtained by Bernard and K\"uhn--Robles--Zeindler, while relying on the same arithmetic inputs. This indicates that optimizing the linear combinations, an approach that has received relatively little attention, has a more pronounced effect than refining the mollifier when it is short. Curiously, our linear combinations provide non-trivial smooth approximations of Siegel's $\mathfrak{f}$-function in the celebrated Riemann--Siegel formula.

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Forward citations

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Reference graph

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Diffusion is a code repair operator and generator Mukul Singh Microsoft Redmond, W A singhmukul@microsoft.com Gust Verbruggen Microsoft Belgium gverbruggen@microsoft.com Vu Le Microsoft Redmond, W A levu@microsoft.com Sumit Gulwani Microsoft Redmond, W A sumitg@microsoft.com Abstract Code diffusion models generate code by iteratively removing noise from t...

work page Pith review arXiv 2020

[2] [2]

We 7 then fine-tune several code generation models on this dataset and evaluate their performance on a repair benchmark containing real human errors

from the diffusion process. We 7 then fine-tune several code generation models on this dataset and evaluate their performance on a repair benchmark containing real human errors. We sample 20K training points. As baselines, we consider generators from existing work, as well as generating data with a large language model (GPT-4o). For Python, we use the pop...

work page 2021

[3] [4]

Diffusion-generated data has more diversity, higher complexity, and more global errors

We show the average distance between token edits (localization), average n-gram similarity between randomly sampled data points (diversity), and average token edit distance between the noisy and correct code (complexity). Diffusion-generated data has more diversity, higher complexity, and more global errors. The diffusion model generates both the code and...

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