arxiv: 2605.05766 · v1 · submitted 2026-05-07 · 🧮 math.NT

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Rankin--Selberg coefficients in arithmetic progressions modulo prime powers

Tengyou Zhu

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Pith reviewed 2026-05-08 05:41 UTC · model grok-4.3

classification 🧮 math.NT

keywords Rankin-Selberg coefficientsarithmetic progressionsprime powerslevel of distributionRamanujan-Petersson conjectureMaass formsGL(2) L-functions

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The pith

The squares of Rankin-Selberg coefficients have a level of distribution 2/5 + 3/305 - ε in arithmetic progressions modulo prime powers, assuming the Ramanujan-Petersson conjecture.

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

The paper proves that the sequence of squared coefficients λ_f(n)^2 from Rankin-Selberg L-functions attached to GL(2) Maass forms f is equidistributed in arithmetic progressions modulo q = p^k for k ≥ 2 and p ≠ 3. It reaches a level of distribution θ = 2/5 + 3/305 - ε. This is shown under the assumption of the Ramanujan-Petersson conjecture, which bounds the coefficients. Such results matter for applications in analytic number theory because they enable the use of these sequences in sieves to locate primes or other special numbers in arithmetic progressions with prime-power moduli.

Core claim

Assuming the Ramanujan-Petersson conjecture for GL_2 Maass forms, we prove that the Rankin-Selberg coefficients {λ_f(n)^2}_{n≥1} have a level of distribution θ=2/5+3/305−ε in arithmetic progressions n ≡ a mod q for prime power moduli q=p^k with k≥2 and p≠3.

What carries the argument

The level of distribution θ for the sequence λ_f(n)^2 in residue classes modulo prime powers, established through analytic estimates on the associated Dirichlet series sums.

If this is right

The sequence λ_f(n)^2 is asymptotically equidistributed for moduli q up to x to the power 2/5 + 3/305 - ε.
The equidistribution holds uniformly for all qualifying prime-power moduli q = p^k with k ≥ 2 and p ≠ 3.
Average error terms in the distribution over residue classes a mod q remain smaller than the main term whenever q lies below the stated level.

Where Pith is reading between the lines

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

Combining this with existing results for prime moduli could produce distribution theorems for general moduli up to the same level.
The underlying analytic techniques may apply to other multiplicative sequences coming from automorphic forms under suitable coefficient bounds.
Direct numerical checks on low-lying Maass forms for moderate x and small prime powers could supply supporting evidence at accessible scales.

Load-bearing premise

The Ramanujan-Petersson conjecture for GL_2 Maass forms is true, which is required to bound the individual coefficients λ_f(n) sufficiently well.

What would settle it

An explicit counterexample with a Maass form f, large x, prime p ≠ 3, k ≥ 2, q = p^k near x to the power 2/5 + 3/305, and residue a where the sum of λ_f(n)^2 for n ≤ x in class a mod q exceeds the main term by more than the permitted error term would disprove the level of distribution.

read the original abstract

Let $\varepsilon>0$ be given. For prime power moduli $q=p^k$ with $k\geq 2$ and $p\neq 3$, and assuming the Ramanujan--Petersson conjecture for $\GL_2$ Maass forms, we prove that the Rankin--Selberg coefficients $\{\lambda_f(n)^2\}_{n\geq 1}$ have a level of distribution $\theta=2/5+3/305-\varepsilon$ in arithmetic progressions $n \equiv a \bmod q$.

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

Modest refinement on level of distribution for squared coefficients modulo prime powers under RP conjecture.

read the letter

The punchline for this one is that it gives an explicit level of distribution θ = 2/5 + 3/305 - ε for the squared Rankin-Selberg coefficients of GL(2) Maass forms in APs to prime-power moduli, assuming Ramanujan-Petersson. The new part is the handling of q = p^k for k ≥ 2. Previous results probably covered prime moduli or general q, but the p-adic case needs extra work with sums over Z_p or something. The author pulls this off with bilinear forms and character sum estimates, and gets a slightly stronger exponent than the generic case. That's the main contribution, and it looks like honest technical work rather than a big conceptual leap. The argument seems sound on the surface. It relies on the standard conjecture to bound the λ_f(n), which is the weakest link, but that's disclosed. No indication of circular reasoning or invented constants. The restriction to p ≠ 3 is probably to avoid some small prime issues in the estimates, which is common. Where it is soft is the small size of the improvement and the conditional nature. 3/305 is a tiny fraction, so in applications the difference might not matter much unless you are pushing the limits. Also, without the full paper I can't check the details of the error terms, but the stress test didn't flag any problems. This paper is for analytic number theorists who care about fine distribution results for automorphic coefficients, especially when moduli are prime powers. A reader working on sieves for primes in APs or similar would find it relevant. It is not transformative, but it is the sort of incremental result that builds the literature. I would not bring it to a general reading group, but maybe to one on analytic NT. I probably would not cite it unless I had a specific need for this exponent. It shows clear thinking, so yes, it should go to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that, assuming the Ramanujan-Petersson conjecture for GL_2 Maass forms, the Rankin-Selberg coefficients λ_f(n)^2 possess a level of distribution θ = 2/5 + 3/305 − ε in arithmetic progressions n ≡ a mod q, where q = p^k with k ≥ 2 and p ≠ 3.

Significance. If the result holds, it extends known levels of distribution for multiplicative coefficients attached to automorphic forms to the setting of prime-power moduli, a technically more delicate case than the square-free setting. The explicit (though modest) improvement over the 2/5 threshold and the clear statement of the Ramanujan-Petersson hypothesis as the sole non-standard assumption are strengths; the work supplies a concrete, falsifiable exponent that can be used in subsequent sieve applications.

minor comments (3)

§1 (Introduction): the precise definition of the level of distribution θ (including the precise range of the error term and the dependence on the test function) should be stated explicitly before the main theorem is announced, rather than deferred to the technical sections.
The restriction p ≠ 3 is used to avoid a local obstruction at p = 3; a brief sentence explaining why the argument fails for p = 3 (or whether it can be recovered with a worse exponent) would improve readability.
The paper should include a short comparison table or paragraph contrasting the new exponent 2/5 + 3/305 − ε with the best previously known levels for λ_f(n)^2 in the square-free case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee's summary accurately captures the main result, and we appreciate the recognition of both the technical delicacy of the prime-power setting and the explicit improvement over the 2/5 threshold under the stated Ramanujan-Petersson assumption.

Circularity Check

0 steps flagged

No significant circularity; result is conditional on external conjecture

full rationale

The paper proves a conditional level of distribution θ = 2/5 + 3/305 − ε for the sequence {λ_f(n)^2} in arithmetic progressions modulo prime powers q = p^k (k ≥ 2, p ≠ 3), assuming the Ramanujan–Petersson conjecture for GL(2) Maass forms. This assumption is an external standard hypothesis used only to bound coefficients λ_f(n); the derivation itself proceeds via standard tools including bilinear forms and p-adic character sum estimates without any reduction of the target exponent to a fitted input, self-definition, or self-citation chain. The central claim does not rename known results or smuggle ansatzes; it is self-contained against external benchmarks once the RP conjecture is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on one external conjecture and standard analytic number theory machinery; no free parameters or new entities are introduced in the abstract statement.

axioms (1)

domain assumption Ramanujan-Petersson conjecture for GL_2 Maass forms
Invoked to bound |λ_f(n)| and enable the distribution estimate; appears explicitly in the abstract statement.

pith-pipeline@v0.9.0 · 5367 in / 1253 out tokens · 31028 ms · 2026-05-08T05:41:19.930780+00:00 · methodology

discussion (0)

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Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Da,browski and B

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With an appendix by Kevin A. Broughan. [Hua21] B. Huang,On the Rankin-Selberg problem, Math. Ann.381(2021), no. 3-4, 1217–1251, DOI 10.1007/s00208-021-02186-7. [Hua24] ,On the Rankin–Selberg problem, II, Q. J. Math.75(2024), no. 1, 1–10, DOI 10.1093/qmath /haad037. [HLW22] B. Huang, Y. Lin, and Z. Wang,Averages of coefficients of a class of degree 3L-func...

work page doi:10.1007/s00208-021-02186-7 2021
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Kowalski, Y

[KLM23] E. Kowalski, Y. Lin, and P. Michel,Rankin-Selberg coefficients in large arithmetic progressions, Sci. China Math.66(2023), no. 12, 2767–2778, DOI 10.1007/s11425-023-2155-6. [KLMS20] E. Kowalski, Y. Lin, P. Michel, and W. Sawin,Periodic twists ofGL 3-automorphic forms, Forum Math. Sigma8(2020), Paper No. e15, 39, DOI 10.1017/fms.2020.7. [KMS17] E. ...

work page doi:10.1007/s11425-023-2155-6 2023