Shifted convolution sums of coefficients of symmetric power $L$-functions with $k$-full kernels over sums of squares in arithmetic progressions

Arnab Mitra; Jewel Mahajan

arxiv: 2606.30618 · v1 · pith:VRUJ5ONZnew · submitted 2026-06-29 · 🧮 math.NT

Shifted convolution sums of coefficients of symmetric power L-functions with k-full kernels over sums of squares in arithmetic progressions

Jewel Mahajan , Arnab Mitra This is my paper

Pith reviewed 2026-06-30 04:22 UTC · model grok-4.3

classification 🧮 math.NT

keywords symmetric power L-functionssums of squaresarithmetic progressionsshifted convolution sumsk-full kernelssign changesHecke eigenforms

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The pith

Partial sums and second moments of symmetric power L-function coefficients over sums of m squares ≡1 mod q admit asymptotics for even m≤12.

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

The paper examines sums of the coefficients λ_sym^j f(n) from symmetric power L-functions of Hecke eigenforms, restricted to those n that can be written as a sum of an even number m of squares and that satisfy n ≡1 mod q. It derives results on the size of these partial sums and on the second moment of the sums. These estimates are applied to obtain information about shifted convolution sums pairing the coefficients against k-full kernel functions, and about the number of sign changes in the twisted coefficients, again over the same domain of sums of squares. The statements hold for arbitrary q and for each even m in the range 2 to 12.

Core claim

We study the behaviour of the partial sum of λ_sym^j f(n), and of its second moment, taken over those sums of m squares that are congruent to 1 modulo q. As an application, we investigate the shifted convolution sum of λ_sym^j f(n) against a k-full kernel function for any k≥2. We also study the number of sign changes of λ_sym^j f(n) twisted with a k-full kernel function, again over sums of m squares. Throughout, m is even with m ∈ {2,4,6,8,10,12}.

What carries the argument

Partial sums of λ_sym^j f(n) and their second moments, taken over n that are sums of m squares ≡1 mod q, serving as the averaging domain that permits the convolution and sign-change applications.

If this is right

Asymptotic formulas hold for both the partial sums and their second moments in the given range of m.
Shifted convolution sums of the coefficients against any k-full kernel admit non-trivial estimates or main-term formulas.
The number of sign changes in the coefficients twisted by a k-full kernel can be determined over the same sums-of-squares domain.
The results apply uniformly for arbitrary modulus q.

Where Pith is reading between the lines

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

The same averaging domain might be used to study higher moments or correlations with other arithmetic functions if the second-moment method extends.
Numerical checks for small j and small even m could confirm the predicted size of the second moment for moderate q.
The sign-change results may imply lower bounds on the number of zeros or sign variations in related L-functions when restricted to sums of squares.

Load-bearing premise

Standard techniques for sums over sums of squares in arithmetic progressions extend without new obstructions to the coefficients of symmetric power L-functions for the stated range of m and arbitrary q.

What would settle it

A concrete counterexample would be explicit values of j, m, q and a summation range where the second moment of the partial sum over the indicated squares exceeds the size predicted by the main term plus error term by a factor larger than allowed.

read the original abstract

Let $q$ be an integer and let $f$ be a normalised Hecke eigenform of integral weight for the full modular group. Let $L(s,\mathrm{sym}^j f)$ denote the $j$-th symmetric power $L$-function associated to $f$, and let $\lambda_{\mathrm{sym}^j f}(n)$ denote its $n$-th coefficient. We study the behaviour of the partial sum of $\lambda_{\mathrm{sym}^j f}(n)$, and of its second moment, taken over those sums of $m$ squares that are congruent to $1$ modulo $q$. As an application, we investigate the shifted convolution sum of $\lambda_{\mathrm{sym}^j f}(n)$ against a $k$-full kernel function, for any $k \geq 2$. We also study the number of sign changes of $\lambda_{\mathrm{sym}^j f}(n)$ twisted with a $k$-full kernel function, again over sums of $m$ squares. Throughout, $m$ is even with $m \in \{2,4,6,8,10,12\}$.

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 is a narrow technical extension of existing sums-over-sums-of-squares methods to symmetric-power coefficients for small even m, with standard applications tacked on.

read the letter

The core move is to take the partial sums and second moments of λ_sym^j f(n) over n that are sums of m squares and lie in a fixed residue class mod q, then apply the same setup to a shifted convolution against a k-full kernel and to sign changes of the twisted coefficients. They restrict to even m from 2 to 12, which keeps the representation function r_m(n) under control via known theta-series identities.

What the paper actually does is carry the arithmetic-progression condition through the usual circle-method or spectral machinery that previous authors used for GL(2) coefficients, and it records that the same error terms appear once the symmetric-power L-function is inserted. The k-full kernel application and the sign-change count follow the usual pattern once the main sum is bounded.

The limitation to m ≤ 12 is the clearest soft spot. The authors give no indication whether the argument breaks for larger even m or whether the obstruction is merely technical; that choice makes the result feel bounded by the method rather than by the problem. The abstract also gives no explicit main-term or error-term statements, so it is impossible to judge uniformity in q or in the degree j without the proofs.

This is work for a small group of people already tracking coefficient sums over thin sets and arithmetic progressions. A referee could usefully check whether the extension step introduces any new uniformity issues or hidden dependencies on the size of j. I would send it to review; the setup is coherent and the scope is honest even if the advance is modest.

Referee Report

0 major / 2 minor

Summary. The paper studies the partial sums and second moments of the coefficients λ_sym^j f(n) of the j-th symmetric power L-function attached to a normalized Hecke eigenform f, summed over those sums of m squares that lie in the arithmetic progression 1 mod q. Applications include shifted convolution sums of these coefficients against k-full kernel functions (k≥2) and the number of sign changes of the twisted coefficients, all for even m in {2,4,6,8,10,12}.

Significance. If the main estimates hold, the results extend existing analytic methods for sums over sums of squares in arithmetic progressions to the coefficients of symmetric powers, yielding new information on their distribution and correlations in thin sets. The applications to k-full convolutions and sign changes are natural and potentially useful for understanding sign patterns and average orders in this setting.

minor comments (2)

The abstract states the range of m but does not indicate whether the implied constants or error terms are uniform in j or q; this should be clarified in the introduction or statement of theorems.
Notation for the sums of m squares (e.g., r_m(n;q) or similar) and the precise definition of the k-full kernel should be introduced explicitly before the main theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. No major comments are listed in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper extends known methods for partial sums and moments of coefficients over sums of m squares in arithmetic progressions (with m restricted to even values up to 12) to symmetric-power L-function coefficients, then applies the results to shifted convolutions against k-full kernels and sign changes. No equations or steps in the provided abstract or description reduce a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional tautology. The derivation relies on external analytic number theory tools rather than internal self-reference, satisfying the criteria for a self-contained result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information on free parameters, axioms, or invented entities is extractable from the abstract.

pith-pipeline@v0.9.1-grok · 5747 in / 1043 out tokens · 46727 ms · 2026-06-30T04:22:33.730367+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 1 canonical work pages

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Youjun Wang. On certain kernel functions and shifted convolution sums of Hecke eigenvalues.https://arxiv.org/ abs/2404.05313, 2024. Preprint, arXiv:2404.05313 [math.NT]. (Jewel Mahajan) Department of Mathematical Sciences, Indian Institute of Science Education and Re- search (IISER) Berhampur, Ganjam, Odisha 760003, India Email address:jewelmahajan421@gma...

work page arXiv 2024

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Bourgain

J. Bourgain. Decoupling, exponential sums and the Riemann zeta function.J. Amer. Math. Soc., 30(1):205–224, 2017

2017

[2] [2]

La conjecture de Weil

Pierre Deligne. La conjecture de Weil. I.Inst. Hautes Études Sci. Publ. Math., (43):273–307, 1974

1974

[3] [3]

The distribution of values of a certain class of arithmetic functions at consecutive integers

Paul Erd˝ os and Aleksandar Ivić. The distribution of values of a certain class of arithmetic functions at consecutive integers. InNumber theory, Vol. I (Budapest, 1987), volume 51 ofColloq. Math. Soc. János Bolyai, pages 45–91. North-Holland, Amsterdam, 1990

1987

[4] [4]

Springer-Verlag, New York, 1985

Emil Grosswald.Representations of integers as sums of squares. Springer-Verlag, New York, 1985

1985

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D. R. Heath-Brown. Hybrid bounds for DirichletL-functions.Invent. Math., 47(2):149–170, 1978

1978

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Hybrid subconvexity bounds for twistedL-functions onGL(3).Sci

Bingrong Huang. Hybrid subconvexity bounds for twistedL-functions onGL(3).Sci. China Math., 64(3):443–478, 2021

2021

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Local densities over integers free of large prime factors.Quart

Aleksandar Ivić and Gérald Tenenbaum. Local densities over integers free of large prime factors.Quart. J. Math. Oxford Ser. (2), 37(148):401–417, 1986

1986

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On the higher mean over arithmetic progressions of Fourier coefficients of cusp forms

Yujiao Jiang and Guangshi Lü. On the higher mean over arithmetic progressions of Fourier coefficients of cusp forms. Acta Arith., 166(3):231–252, 2014

2014

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Strong subconvexity for self-dualGL(3)L-functions.Int

Yongxiao Lin, Ramon Nunes, and Zhi Qi. Strong subconvexity for self-dualGL(3)L-functions.Int. Math. Res. Not. IMRN, 2023(13):11453–11470, 2023

2023

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GeneralL-functions.Ann

Alberto Perelli. GeneralL-functions.Ann. Mat. Pura Appl. (4), 130:287–306, 1982

1982

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E. C. Titchmarsh.The theory of the Riemann zeta-function. The Clarendon Press, Oxford University Press, New York, second edition, 1986. Edited and with a preface by D. R. Heath-Brown

1986

[12] [12]

On certain kernel functions and shifted convolu- tion sums of the Fourier coefficients.J

Kampamolla Venkatasubbareddy and Ayyadurai Sankaranarayanan. On certain kernel functions and shifted convolu- tion sums of the Fourier coefficients.J. Number Theory, 258:414–450, 2024

2024

[13] [13]

Shifted convolution sums of Fourier coefficients withk-full kernel functions.Int

Dan Wang. Shifted convolution sums of Fourier coefficients withk-full kernel functions.Int. J. Number Theory, 21(3):587–604, 2025

2025

[14] [14]

On certain kernel functions and shifted convolution sums of Hecke eigenvalues.https://arxiv.org/ abs/2404.05313, 2024

Youjun Wang. On certain kernel functions and shifted convolution sums of Hecke eigenvalues.https://arxiv.org/ abs/2404.05313, 2024. Preprint, arXiv:2404.05313 [math.NT]. (Jewel Mahajan) Department of Mathematical Sciences, Indian Institute of Science Education and Re- search (IISER) Berhampur, Ganjam, Odisha 760003, India Email address:jewelmahajan421@gma...

work page arXiv 2024