arxiv: 2605.09322 · v1 · submitted 2026-05-10 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

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

Rongjie Yin, Tengyou Zhu

Pith reviewed 2026-05-12 02:19 UTC · model grok-4.3

classification 🧮 math.NT

keywords exponent of distributionHecke eigenvaluesarithmetic progressionscusp formsconvolutionssmooth moduliGL(2) coefficients

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The pith

The convolution of Hecke eigenvalues with the constant function distributes with exponent 1/2 + 1/70 in arithmetic progressions to square-free smooth moduli.

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

The paper proves an explicit lower bound on the exponent of distribution for the sequence formed by convolving the Hecke eigenvalues of a holomorphic cusp form with the constant function 1. This exponent reaches 1/2 plus 1/70 precisely when the modulus is square-free and its prime factors are all sufficiently small. A reader cares because such distribution bounds control the error terms when summing the sequence over residue classes, which in turn feeds into sieve applications and estimates for the number of integers represented by the sequence in short intervals. The result is an extension of earlier work on the distribution of GL(2) coefficients.

Core claim

Let (λ_f(n))_{n≥1} be the Hecke eigenvalues of a holomorphic cusp form f. We prove that the exponent of distribution of λ_f * 1 in arithmetic progressions is as large as 1/2 + 1/70 when the modulus q is square-free and has only sufficiently small prime factors.

What carries the argument

The exponent of distribution for the convolution λ_f * 1, which bounds the maximal δ such that the partial sums of the sequence in arithmetic progressions modulo q have error O(x / q^δ) for x up to the length of the sum; the proof establishes this δ = 1/2 + 1/70 under the stated restrictions on q.

If this is right

The bound improves the range of moduli for which sums of λ_f * 1 over residue classes can be evaluated with a power-saving error.
It supplies a concrete saving beyond the square-root barrier for this particular convolution under the smoothness condition on q.
The result directly strengthens applications of the sequence to problems that require uniform distribution in arithmetic progressions, such as counting integers free of large prime factors in certain congruence classes.

Where Pith is reading between the lines

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

Removing the small-prime-factor restriction would immediately enlarge the set of moduli to which the bound applies, but the paper gives no indication that the current method can achieve this.
The same analytic machinery might be tested on convolutions involving Maass forms or higher-rank automorphic forms to see whether an analogous saving appears.
Direct computation of the distribution error for small q and small cusp forms could check whether 1/70 is close to the optimal constant under the given hypotheses.

Load-bearing premise

The modulus q must be square-free and composed only of sufficiently small prime factors.

What would settle it

A numerical or analytic computation showing that for some square-free modulus q containing a prime factor larger than the allowed threshold, the distribution error term for λ_f * 1 exceeds what the claimed exponent permits.

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 $\frac{1}{2}+\frac{1}{70}$ when the modulus $q$ is square-free and has only sufficiently small prime factors.

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

Yin and Zhu improve the distribution exponent for λ_f * 1 to 1/2 + 1/70, but only for square-free moduli with small prime factors.

read the letter

The main takeaway is that Yin and Zhu have shown the convolution λ_f * 1 has an exponent of distribution reaching 1/2 + 1/70 in arithmetic progressions, provided the modulus q is square-free and composed of sufficiently small primes. This is new as a specific improvement for this particular sequence under those constraints. They adapt techniques from previous studies on the distribution of Hecke eigenvalues to obtain the extra 1/70. The paper does well by delivering a clean statement of the theorem and by being upfront about the conditions needed for the result to hold. The approach appears to rely on combining spectral estimates for GL(2) forms with large sieve inequalities, which is a standard but effective route for these problems. Credit goes to the authors for making the gain explicit and for restricting the claim to where it works. The soft spots are the limited scope and the size of the advance. The restriction to square-free q with small prime factors means the result applies only in special cases, which reduces its immediate usefulness for many sieve applications that require more general moduli. The 1/70 improvement is positive but small, and it does not break through to a major new range. Without the full manuscript details, I cannot verify the precise error term management or any auxiliary estimates, though nothing in the abstract suggests a problem. This kind of paper is for researchers in analytic number theory who focus on arithmetic progressions and automorphic L-functions. A reader who needs better distribution results for convolutions involving cusp form coefficients might find the method adaptable. It deserves serious refereeing because the statement is precise and the topic is relevant to ongoing work in the area. I recommend sending it out for peer review.

Referee Report

0 major / 1 minor

Summary. The paper proves that the exponent of distribution of λ_f * 1 (where λ_f are the Hecke eigenvalues of a holomorphic cusp form f) in arithmetic progressions is at least 1/2 + 1/70, provided the modulus q is square-free and has only sufficiently small prime factors.

Significance. If the proof is correct, this establishes a concrete improvement over the square-root barrier for the distribution of this GL(2) convolution in APs, under the standard restriction to smooth square-free moduli. Such exponents are load-bearing for applications in sieve theory and the study of automorphic L-functions; the explicit constant 1/70 is a modest but verifiable gain that could be useful in conditional results.

minor comments (1)

[Abstract] The abstract and introduction should clarify the dependence of the 'sufficiently small' prime-factor bound on the level of f and on the implied constants, to make the range of applicability fully explicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript 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 theorem proving an exponent of distribution 1/2 + 1/70 for λ_f * 1 in APs restricted to square-free smooth moduli q. The abstract and context present this as a derived result from analytic estimates (spectral theory, large sieve), with the modulus restriction stated explicitly as a hypothesis rather than derived circularly. No load-bearing step reduces the claimed exponent to a fitted input, self-definition, or unverified self-citation chain; the derivation chain remains independent of the target result by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard properties of holomorphic cusp forms (Deligne bound, Rankin-Selberg L-functions) and analytic number theory tools such as the circle method or spectral large sieve, plus the assumption that q is smooth and square-free. No new entities are introduced.

axioms (2)

standard math Hecke eigenvalues satisfy the Ramanujan conjecture (Deligne bound)
Invoked implicitly for bounding the coefficients in the distribution estimate.
domain assumption Standard estimates for the divisor function and bilinear forms in arithmetic progressions
Required to handle the convolution λ_f * 1.

pith-pipeline@v0.9.0 · 5353 in / 1333 out tokens · 33774 ms · 2026-05-12T02:19:05.118716+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
We prove that the exponent of distribution of λ_f * 1 ... is as large as 1/2 + 1/70 when the modulus q is square-free and has only sufficiently small prime factors.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Bilinear forms ... X_m X_l α_m β_l Kl_3(aml;q) ≪ ... (Thm 2.1)

Reference graph

Works this paper leans on

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