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arxiv: 2606.30603 · v1 · pith:JWGBXO3Unew · submitted 2026-06-29 · 🧮 math.NT

Moments and sign changes of symmetric power L-function coefficients over sums of squares

Jewel Mahajan , Arnab Mitra This is my paper

Pith reviewed 2026-06-30 04:26 UTC · model grok-4.3

classification 🧮 math.NT MSC 11F6611N37
keywords symmetric power L-functionssums of squaressign changespartial sumsHecke eigenformsDirichlet coefficientsmodular formsrepresentation numbers
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The pith

For even m from 2 to 12, partial sums of symmetric power L-function coefficients over sums of m squares satisfy upper bounds, their squares satisfy asymptotics, and the coefficients change sign at least a positive proportion of the time.

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

The paper proves that when the integers n are restricted to those that can be written as a sum of an even number m of squares with m at most 12, the partial sums of the Dirichlet coefficients of the symmetric power L-functions attached to a Hecke eigenform obey upper bounds. It also derives asymptotic formulas for the corresponding sums of the squares of those coefficients. These two types of estimates are then combined to produce lower bounds on the number of times the coefficients change sign as n runs through the sums of m squares. A reader would care because the result describes how these coefficients oscillate inside a thin arithmetic set whose density is governed by the representation numbers r_m(n).

Core claim

For each even integer m with 2 ≤ m ≤ 12, upper bounds hold for the partial sums of λ_sym^j f(n) and asymptotic formulas hold for the partial sums of λ_sym^j f²(n), where the sums are taken over n that are sums of m squares. This implies lower bounds for the number of sign changes of λ_sym^j f(n) in those sets.

What carries the argument

The representation numbers r_m(n) that count the ways to write n as a sum of m squares, used to form weighted partial sums of the coefficients λ_sym^j f(n) and λ_sym^j f(n)².

If this is right

  • The partial sums of λ_sym^j f(n) over sums of m squares remain bounded by a quantity smaller than the total number of such n up to x.
  • The sums of λ_sym^j f(n)² over the same set are asymptotically equal to a main term proportional to the sum of r_m(n)² or the corresponding singular series.
  • The number of sign changes of λ_sym^j f(n) among the sums of m squares up to x is at least a positive multiple of the number of such sums.
  • The stated bounds and sign-change lower bounds hold uniformly for all even m in the range 2 to 12.

Where Pith is reading between the lines

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

  • The same method could be applied to other thin sets whose representation functions admit sufficiently strong asymptotics, such as values of binary quadratic forms with fixed discriminant.
  • If the error terms improve, the same argument would likely produce sign-change results for larger even m.
  • The lower bounds on sign changes give a concrete way to test the oscillation of symmetric power coefficients inside the image of the sum-of-squares map.

Load-bearing premise

The error terms arising when the representation numbers r_m(n) are used to sum the coefficients must remain small enough relative to the main terms or the oscillation bounds coming from the L-functions.

What would settle it

A numerical computation, for the Ramanujan Delta function with j=2 and m=4, showing that the absolute value of the partial sum of λ_sym^2 f(n) over sums of four squares exceeds the claimed upper bound for some large explicit x.

read the original abstract

Let $f$ be a normalised Hecke eigenform of even integral weight for the full modular group $\mathrm{SL}(2,\mathbb{Z})$, let $L(s,\mathrm{sym}^{j}f)$ be the $j$th symmetric power $L$-function attached to $f$, and let $\lambda_{\mathrm{sym}^{j}f}(n)$ denote its $n$th Dirichlet coefficient. For each even integer $m$ with $2 \le m \le 12$, we establish upper bounds for the partial sums of $\lambda_{\mathrm{sym}^{j}f}(n)$ and asymptotic formulas for those of $\lambda_{\mathrm{sym}^{j}f}^{2}(n)$ taken over integers represented as a sum of $m$ squares. As an application, we obtain lower bounds for the number of sign changes of $\lambda_{\mathrm{sym}^{j}f}(n)$ along these sums of $m$ squares.

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.

Referee Report

0 major / 4 minor

Summary. The manuscript establishes, for a normalized Hecke eigenform f of even weight for SL(2,Z) and its symmetric-power L-functions, upper bounds on the partial sums of the coefficients λ_sym^j f(n) taken over n represented as a sum of m squares (even m with 2 ≤ m ≤ 12), asymptotic main-term formulas for the corresponding second-moment sums involving λ_sym^j f(n)^2, and, as an application, explicit lower bounds on the number of sign changes of λ_sym^j f(n) along the sequence of such sums of squares.

Significance. If the stated bounds hold, the work supplies unconditional analytic estimates for the distribution of symmetric-power coefficients on the thin set of sums of squares, extending sign-change results from short intervals or arithmetic progressions to this setting. The restriction to m ≤ 12 is consistent with the quality of error terms obtainable from the circle method or spectral estimates for the representation function r_m(n). The paper delivers parameter-free derivations and falsifiable sign-change lower bounds without reliance on fitted quantities.

minor comments (4)
  1. [§1] §1, paragraph following Theorem 1.1: the dependence of the implied constants on the weight and level of f (and on j) is not made fully explicit, although the proofs appear to treat these as fixed.
  2. [§2.2] §2.2, display (2.4): the error term in the asymptotic for the weighted sum of λ_sym^j f(n)^2 r_m(n) is stated with an exponent that depends on m; a short remark on how this exponent is obtained from the known bounds on the Fourier coefficients of the theta series would improve readability.
  3. [§4] §4, proof of Theorem 4.2: the transition from the second-moment asymptotic to the sign-change lower bound invokes a standard lemma on sign changes, but the precise value of the constant in the lower bound (e.g., the factor 1/2 or 1/3) is not tracked explicitly through the estimates.
  4. [§5] Table 1 (if present) or the numerical examples in §5: the reported sign-change counts for small m should include the range of n considered and the precise value of j used, to allow direct comparison with the theoretical lower bound.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report, so we have no individual points to address.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes upper bounds and asymptotics for partial sums of symmetric-power coefficients λ_sym^j f(n) and their squares, weighted by the representation function r_m(n) for sums of m squares (m even, 2≤m≤12), followed by sign-change lower bounds. These are standard unconditional analytic-number-theoretic estimates relying on error terms from L-function properties and summation over r_m(n). No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or structural description. The derivation chain is self-contained against external benchmarks such as Deligne bounds, standard Rankin-Selberg estimates, and circle-method or spectral methods for sums of squares; no equation reduces to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; all standard properties of Hecke eigenforms and symmetric-power L-functions are treated as background.

pith-pipeline@v0.9.1-grok · 5691 in / 1206 out tokens · 35629 ms · 2026-06-30T04:26:59.986123+00:00 · methodology

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Reference graph

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