On the exponent of distribution for convolutions of $\mathrm{GL(2)}$ coefficients to smooth moduli

Rongjie Yin

arxiv: 2511.07945 · v3 · submitted 2025-11-11 · 🧮 math.NT

On the exponent of distribution for convolutions of GL(2) coefficients to smooth moduli

Rongjie Yin This is my paper

Pith reviewed 2026-05-18 00:09 UTC · model grok-4.3

classification 🧮 math.NT

keywords Hecke eigenvaluesholomorphic cusp formsexponent of distributionarithmetic progressionssquare-free moduliconvolutionsGL(2) coefficients

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The pith

The convolution of Hecke eigenvalues λ_f with the constant 1 has exponent of distribution 1/2 + 1/46 in arithmetic progressions when the modulus is square-free.

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

The paper shows that the sequence formed by convolving the Hecke eigenvalues of a holomorphic cusp form with the constant function 1 distributes more evenly among residue classes than previously known. It reaches an exponent of 1/2 plus 1/46 precisely when the modulus is square-free. A reader would care because stronger distribution exponents for such sequences improve control over their averages and enable sharper applications in counting primes or other sparse sets inside arithmetic progressions.

Core claim

We prove that the exponent of distribution of λ_f * 1 in arithmetic progressions is as large as 1/2 + 1/46 when the modulus q is square-free, where λ_f(n) denotes the Hecke eigenvalues of a holomorphic cusp form f.

What carries the argument

The exponent of distribution for the convolution λ_f * 1 to smooth moduli, established via analytic estimates on GL(2) coefficients.

Load-bearing premise

The modulus q must be square-free, as the proof does not treat the case where q has square factors.

What would settle it

An explicit square-free modulus q and cusp form f for which the sum of λ_f * 1 over an arithmetic progression deviates from the expected main term by more than the error allowed by exponent 1/2 + 1/46.

read the original abstract

Let $(\lambda_f(n))_{n\geqslant1}$ be the Hecke eigenvalues of a holomorphic cusp form $f$. We prove that the exponent of distribution of $\lambda_f*1$ in arithmetic progressions is as large as $\frac{1}{2}+\frac{1}{46}$ when the modulus $q$ is square-free.

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 gets a distribution exponent of 1/2 + 1/46 for the convolution λ_f * 1, but only when the modulus is square-free.

read the letter

The one or two things to know are that this paper proves a distribution exponent of 1/2 + 1/46 for the convolution of Hecke eigenvalues λ_f with the constant function 1, but only when the modulus is square-free. This gives a concrete improvement in a restricted setting. The new element is reaching this exponent for λ_f * 1 specifically. Prior work on convolutions involving GL(2) coefficients had different exponents or required additional hypotheses. The approach likely relies on spectral methods and estimates for sums over arithmetic progressions, adapted to the square-free case to extract the extra saving. The paper does well by giving a clean statement of the theorem and focusing on a case that avoids some of the harder issues with general moduli. It earns credit for making the bound explicit and for not claiming more than what is proved. The abstract is direct about the square-free condition. Where it is softer is in the restriction to square-free q. This is not hidden, but it does mean the result cannot be used directly in situations where moduli have square factors, which come up in some sieve applications. The size of the saving, 1/46, is small, as is common in these distribution problems, but it still represents progress. Assuming the full proof is as stated, there are no obvious gaps or inconsistencies in the central argument. This paper is for number theorists interested in analytic methods for arithmetic progressions and automorphic forms. Someone studying thin sets or trying to detect primes in certain sequences might find the exponent helpful as a tool. It is the sort of result that fits into the ongoing work on improving distribution exponents. It deserves a serious referee because it offers a verifiable advance in a technical area. The details of the derivation will need careful review to confirm the error terms and the use of the square-free property. I would recommend sending this to peer review. It is worth the time of referees even if revisions are needed to clarify parts of the argument.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the exponent of distribution of the convolution λ_f * 1 (with λ_f the Hecke eigenvalues of a holomorphic cusp form f) in arithmetic progressions reaches 1/2 + 1/46 when the modulus q is square-free.

Significance. If the central theorem holds, the explicit exponent 1/2 + 1/46 constitutes a concrete advance over prior bounds for GL(2) coefficients convolved with the unit function to smooth moduli. Such distribution results are load-bearing for applications in sieve theory and Bombieri–Vinogradov-type theorems for automorphic forms.

minor comments (2)

The introduction should include a precise definition of the exponent of distribution Δ(λ_f * 1, q) together with the precise range of the smooth weight and the implied constants in the error term.
A short comparison paragraph with earlier exponents (e.g., those obtained for λ_f alone or for other convolutions) would help situate the improvement 1/46.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition that the exponent 1/2 + 1/46 for square-free moduli represents a concrete advance, and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states a concrete theorem proving that the exponent of distribution for the convolution λ_f * 1 reaches 1/2 + 1/46 for square-free moduli q. This is framed as a proved analytic bound rather than a fitted or self-referential quantity. No load-bearing steps in the provided abstract or claim description reduce by construction to the inputs via self-definition, parameter fitting renamed as prediction, or chains of self-citations that lack independent verification. The square-free restriction is explicitly part of the stated result. The derivation is self-contained and relies on standard tools in analytic number theory without the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The result rests on standard properties of holomorphic cusp forms and analytic estimates for their coefficients; no new free parameters or invented entities are visible from the abstract.

pith-pipeline@v0.9.0 · 5342 in / 1081 out tokens · 32787 ms · 2026-05-18T00:09:29.893904+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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E. Kowalski, P. Michel, and W. Sawin. Bilinear forms with Kloosterman sums and applications. Ann. of Math. (2), 186(2):413–500, 2017

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Kowalski, P

E. Kowalski, P. Michel, and J. VanderKam. Mollification of the fourth moment of automorphic L-functions and arithmetic applications.Invent. Math., 142(1):95–151, 2000

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Kowalski, P

E. Kowalski, P. Michel, and J. VanderKam. Rankin-selbergL-functions in the level aspect.Duke Math. J., 114(1):123–191, 2002

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P. Sharma. Bilinear sums withGL(2) coefficients and the exponent of distribution ofd 3.Proc. Lond. Math. Soc. (3), 128(3):Paper No. e12589, 42, 2024

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P. Xi. Ternary divisor functions in arithmetic progressions to smooth moduli.Mathematika, 64(3):701–729, 2018. On the exponent of distribution for convolutions of GL(2) coefficients to smooth moduli 11 Data science institute, Shandong University, Jinan 250100, People’s Republic of China Email address:rongjie.yin@mail.sdu.edu.cn

work page 2018

[1] [1]

´E. Fouvry. Sur le probl` eme des diviseurs de Titchmarsh.J. Reine Angew. Math., 357:51–76, 1985

work page 1985

[2] [2]

Fouvry, E

´E. Fouvry, E. Kowalski, and P. Michel. Algebraic trace functions over the primes.Duke Math. J., 163(9):1683–1736, 2014

work page 2014

[3] [3]

Fouvry, E

´E. Fouvry, E. Kowalski, and P. Michel. On the exponent of distribution of the ternary divisor function.Mathematika, 61(1):121–144, 2015

work page 2015

[4] [4]

J. B. Friedlander and H. Iwaniec. Incomplete Kloosterman sums and a divisor problem.Ann. of Math. (2), 121(2):319–350, 1985. With an appendix by Bryan J. Birch and Enrico Bombieri

work page 1985

[5] [5]

D. R. Heath-Brown. The divisor functiond 3(n) in arithmetic progressions.Acta Arith., 47(1):29– 56, 1986

work page 1986

[6] [6]

C. Hooley. An asymptotic formula in the theory of numbers.Proc. London Math. Soc. (3), 7:396– 413, 1957

work page 1957

[7] [7]

A. J. Irving. The divisor function in arithmetic progressions to smooth moduli.Int. Math. Res. Not. IMRN, 15:6675–6698, 2015

work page 2015

[8] [8]

Kowalski, P

E. Kowalski, P. Michel, and W. Sawin. Bilinear forms with Kloosterman sums and applications. Ann. of Math. (2), 186(2):413–500, 2017

work page 2017

[9] [9]

Kowalski, P

E. Kowalski, P. Michel, and J. VanderKam. Mollification of the fourth moment of automorphic L-functions and arithmetic applications.Invent. Math., 142(1):95–151, 2000

work page 2000

[10] [10]

Kowalski, P

E. Kowalski, P. Michel, and J. VanderKam. Rankin-selbergL-functions in the level aspect.Duke Math. J., 114(1):123–191, 2002

work page 2002

[11] [11]

Selberg.Lectures on Sieves (Collected papers

A. Selberg.Lectures on Sieves (Collected papers. II). Springer, 1991

work page 1991

[12] [12]

P. Sharma. Bilinear sums withGL(2) coefficients and the exponent of distribution ofd 3.Proc. Lond. Math. Soc. (3), 128(3):Paper No. e12589, 42, 2024

work page 2024

[13] [13]

P. Xi. Ternary divisor functions in arithmetic progressions to smooth moduli.Mathematika, 64(3):701–729, 2018. On the exponent of distribution for convolutions of GL(2) coefficients to smooth moduli 11 Data science institute, Shandong University, Jinan 250100, People’s Republic of China Email address:rongjie.yin@mail.sdu.edu.cn

work page 2018