On a level analog of Selberg's result on $S(t)$

Hui Wang; Qingfeng Sun

arxiv: 2407.14867 · v3 · submitted 2024-07-20 · 🧮 math.NT

On a level analog of Selberg's result on S(t)

Qingfeng Sun , Hui Wang This is my paper

Pith reviewed 2026-05-23 22:34 UTC · model grok-4.3

classification 🧮 math.NT

keywords S(t,f)momentslevel aspectSelbergHecke cusp formscentral limit theoremzero-density estimatesL-functions

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The pith

The moments of S(t,f) for L-functions of prime-level cusp forms admit an unconditional asymptotic formula.

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

The paper establishes an asymptotic formula for the moments of S(t,f), defined as the normalized argument of L(1/2 + it, f) where f is a holomorphic Hecke cusp form of weight 2 and prime level q. This provides a level-aspect version of Selberg's classical result on the moments of S(t) attached to the Riemann zeta function. The result matters to a reader interested in the statistical behavior of L-functions because the moments control how the phase fluctuates along the critical line as the level varies. The authors reach the formula by approximating S(t,f) with a truncated Dirichlet series and combining it with a weighted zero-density estimate for the family.

Core claim

We establish an unconditional asymptotic formula for the moments of S(t,f)=π^{-1} arg L(1/2+it, f), where f is a holomorphic Hecke cusp form of weight 2 and prime level q. This provides a level aspect analogue of Selberg's classical work on S(t). As a consequence, we derive a weighted central limit theorem for the distribution of S(t,f) normalized by sqrt(log log q). To this end, we develop a precise approximation for S(t,f) via a truncated Dirichlet series and employ a weighted zero-density estimate for the corresponding family of L-functions.

What carries the argument

Truncated Dirichlet series approximation for S(t,f) combined with a weighted zero-density estimate for the family of L-functions.

Load-bearing premise

The weighted zero-density estimate for the family of L-functions holds and combines with the truncated Dirichlet series approximation to yield the moment asymptotics without further hypotheses.

What would settle it

Compute the second or fourth moment of S(t,f) numerically over a range of t for a sequence of increasing prime levels q and check whether the values match the explicit asymptotic predicted by the formula.

read the original abstract

Let $S(t,f)=\pi^{-1}\arg L(1/2+it, f)$, where $f$ is a holomorphic Hecke cusp form of weight $2$ and prime level $q$. In this paper, we establish an unconditional asymptotic formula for the moments of $S(t,f)$, providing a level aspect analogue of Selberg's classical work on $S(t)$. As a consequence, we derive a weighted central limit theorem for the distribution of $S(t,f)$ normalized by $\sqrt{\log\log q}$. To this end, we develop a precise approximation for $S(t,f)$ via a truncated Dirichlet series and employ a weighted zero-density estimate for the corresponding family of $L$-functions.

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

This paper claims an unconditional level-aspect analogue of Selberg's moment results for S(t) on the family of L-functions attached to weight-2 Hecke forms of prime level q, using a truncated Dirichlet series and a weighted zero-density estimate.

read the letter

The core contribution is a level-aspect version of Selberg's asymptotics for the moments of S(t), now for S(t,f) where f runs over holomorphic Hecke cusp forms of weight 2 and prime level q. They also extract a weighted central limit theorem with normalization sqrt(log log q). The approach relies on approximating S(t,f) by a truncated Dirichlet series and pairing it with a weighted zero-density estimate developed for this specific family. That combination is what lets them claim the results are unconditional. The setup follows the classical pattern from Selberg but adapts the tools to the level aspect with prime q, which is a reasonable technical step. The citation pattern is standard and draws on existing zero-density work without obvious omissions. The main uncertainty sits in the weighted zero-density estimate itself. It needs to suppress the right range of zeros and deliver error terms small enough to not disturb the moment asymptotics uniformly in the order k. If the weights achieve that without extra hypotheses, the argument goes through; otherwise the unconditional claim weakens. The abstract presents the results as unconditional, so the paper stands or falls on whether that estimate is proved cleanly. This is the sort of targeted extension that specialists in analytic number theory working on L-functions in families would find useful to see in print. It is not a broad breakthrough but it fills a specific gap with concrete estimates. A serious referee should be able to check the zero-density section in detail and confirm the error control. I would send it to peer review rather than desk reject.

Referee Report

2 major / 1 minor

Summary. The paper claims to establish an unconditional asymptotic formula for the moments of S(t,f) = π^{-1} arg L(1/2 + it, f) for holomorphic Hecke cusp forms f of weight 2 and prime level q. This is presented as a level-aspect analogue of Selberg's classical result on S(t). As a consequence, a weighted central limit theorem is derived for the distribution of S(t,f) normalized by √(log log q). The proof relies on a truncated Dirichlet series approximation for S(t,f) combined with a weighted zero-density estimate for the associated family of L-functions.

Significance. If the central claims hold, the work provides a meaningful extension of Selberg's theorem to the level aspect for a family of L-functions, contributing to the understanding of value distributions in families. The development of the weighted zero-density estimate is a technical strength that could have independent interest, and the unconditional nature (via standard tools) adds to its value if the error control is verified.

major comments (2)

[The section developing and applying the weighted zero-density estimate (combined with the truncated Dirichlet series)] The weighted zero-density estimate (developed for the family of L-functions attached to weight-2 holomorphic Hecke forms of prime level q): the manuscript must explicitly verify that this estimate is unconditional, that the weights suppress contributions from zeros with |Re(ρ) - 1/2| ≪ 1/log(q(1+|t|)), and that the resulting error term is o((log log q)^{k/2}) uniformly in the moment order k. This is load-bearing for the unconditional moment asymptotics and weighted CLT claimed in the abstract.
[The section on the approximation for S(t,f) and its use in moment calculations] The combination of the truncated Dirichlet series approximation with the zero-density estimate: the error analysis must confirm that no auxiliary hypotheses are needed for the moment asymptotics to hold in the stated range, as this step directly determines whether the results follow unconditionally.

minor comments (1)

[Introduction and statement of main theorems] Clarify the precise range of k for which the asymptotic formula holds and ensure uniformity statements are explicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the points requiring greater explicitness. The concerns raised pertain to the clarity of the unconditional nature of the estimates and the error control; both can be addressed by expanding the relevant sections with additional verifications and explicit bounds. We outline our point-by-point responses below.

read point-by-point responses

Referee: [The section developing and applying the weighted zero-density estimate (combined with the truncated Dirichlet series)] The weighted zero-density estimate (developed for the family of L-functions attached to weight-2 holomorphic Hecke forms of prime level q): the manuscript must explicitly verify that this estimate is unconditional, that the weights suppress contributions from zeros with |Re(ρ) - 1/2| ≪ 1/log(q(1+|t|)), and that the resulting error term is o((log log q)^{k/2}) uniformly in the moment order k. This is load-bearing for the unconditional moment asymptotics and weighted CLT claimed in the abstract.

Authors: We agree that explicit verification strengthens the presentation. In the revised manuscript we will insert a dedicated paragraph (or short subsection) immediately after the statement of the weighted zero-density estimate. There we will: (i) recall that the estimate follows from the standard zero-density theorem of Heath-Brown type applied to the family, with no appeal to GRH or other auxiliary hypotheses; (ii) exhibit the explicit weight function and show by direct estimation that the contribution of any zero with |Re(ρ)−1/2| ≫ 1/log(q(1+|t|)) is absorbed into the error term; (iii) track the resulting error through the moment calculation and confirm it is o((log log q)^{k/2}) uniformly for the range of k appearing in the asymptotic formula and the weighted CLT. These additions will be purely expository and will not alter the existing proofs. revision: yes
Referee: [The section on the approximation for S(t,f) and its use in moment calculations] The combination of the truncated Dirichlet series approximation with the zero-density estimate: the error analysis must confirm that no auxiliary hypotheses are needed for the moment asymptotics to hold in the stated range, as this step directly determines whether the results follow unconditionally.

Authors: We will expand the error analysis in the section combining the truncated Dirichlet series with the weighted zero-density estimate. The revised text will contain a self-contained paragraph that assembles all error terms arising from truncation, from the zero-density bound, and from the approximation of the argument by the Dirichlet series. Each term will be bounded explicitly, showing that the total error remains smaller than the main term throughout the range of t and k under consideration, using only the unconditional tools already invoked in the paper. No additional hypotheses will be introduced or required. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses newly developed estimates independent of the target result.

full rationale

The paper develops a weighted zero-density estimate for the family of L-functions attached to weight-2 holomorphic Hecke forms of prime level q, then combines it with a truncated Dirichlet series approximation to obtain the moment asymptotics for S(t,f). These steps are presented as original contributions within the manuscript and do not reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations. The result is claimed unconditional precisely because the zero-density estimate is proved in the paper rather than assumed or imported from prior self-work. No quoted equation or step exhibits the patterns of self-definitional reduction or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable.

pith-pipeline@v0.9.0 · 5641 in / 1084 out tokens · 24116 ms · 2026-05-23T22:34:01.184572+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Billingsley, Probability and measure

P. Billingsley, Probability and measure . (English summary) Third edition. Wiley Series in Probabil ity and Mathematical Statistics. A Wiley-Interscience Publicati on. John Wiley & Sons, Inc., New York, 1995. xiv+593 pp

work page 1995

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Davenport, Multiplicative number theory

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work page 1980

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Gradshteyn and I

I. Gradshteyn and I. Ryzhik, Table of integrals, series, and products. Translated from the Russian. Translation edited and with a preface by Alan Jeﬀrey and Daniel Zwillinge r. With one CD-ROM (Windows, Macintosh and UNIX). Seventh edition. Elsevier/Academic Press, Amst erdam, 2007. xlviii+1171 pp

work page 2007

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Hough, Zero-density estimate for modular form L-functions in weight aspect

B. Hough, Zero-density estimate for modular form L-functions in weight aspect . Acta Arith. 154 (2012), no. 2, 187–216. 19

work page 2012

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Iwaniec and E

H. Iwaniec and E. Kowalski, Analytic number theory. American Mathematical Society Colloquium Publica- tions, 53. American Mathematical Society, Providence, RI, 2004. xii+615 pp

work page 2004

[7] [7]

Iwaniec, W

H. Iwaniec, W. Luo and P. Sarnak, Low lying zeros of families of L-functions. Inst. Hautes ´Etudes Sci. Publ. Math. No. 91 (2000), 55–131

work page 2000

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Kowalski and P

E. Kowalski and P. Michel, The analytic rank of J0pqq and zeros of automorphic L-functions. Duke Math. J. 100 (1999), no. 3, 503–542

work page 1999

[9] [9]

Kowalski and P

E. Kowalski and P. Michel, Explicit upper bound for the (analytic) rank of J0pqq. Israel J. Math. 120 (2000), part A, 179–204

work page 2000

[10] [10]

S. C. Liu and S. Liu, A GL3 analog of Selberg’s result on Sptq. Ramanujan J. 56 (2021), no. 1, 163–181

work page 2021

[11] [11]

S. C. Liu and J. Shim, Moments of Spt, fq associated with holomorphic Hecke cusp forms . Taiwanese J. Math. 26 (2022), no. 3, 463–482

work page 2022

[12] [12]

Selberg, On the remainder in the formula for N pT q, the number of zeros of ζpsq in the strip 0 ă t ă T

A. Selberg, On the remainder in the formula for N pT q, the number of zeros of ζpsq in the strip 0 ă t ă T . Avh. Norske Vid.-Akad. Oslo I 1944 (1944), no. 1, 27 pp

work page 1944

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Selberg, Contributions to the theory of Dirichlet’s L-functions

A. Selberg, Contributions to the theory of Dirichlet’s L-functions . Skr. Norske Vid.-Akad. Oslo I 1946 (1946), no. 3, 62 pp

work page 1946

[14] [14]

Selberg, Contributions to the theory of the Riemann zeta-function

A. Selberg, Contributions to the theory of the Riemann zeta-function . Arch. Math. Naturvid. 48 (1946), no. 5, 89–155

work page 1946

[15] [15]

Shimura, Introduction to the arithmetic theory of automorphic funct ions

G. Shimura, Introduction to the arithmetic theory of automorphic funct ions. Kanˆ o Memorial Lectures, No

work page

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Publications of the Mathematical Society of Japan, No. 11 . Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, NJ, 1971. xiv+267 pp. School of Mathematics and Statistics, Shandong University , Weihai, Weihai, Shandong 264209, China Email address : qfsun@sdu.edu.cn School of Mathematics, Shandong University, Jinan, Shando ng 250100, C...

work page 1971