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arxiv: 2605.20707 · v1 · pith:5A2WEU2Dnew · submitted 2026-05-20 · 🧮 math.NT

Limiting Distribution and Rate of Convergence for GL(3) Fourier Coefficients

Zongqi Yu This is my paper

Pith reviewed 2026-05-21 02:49 UTC · model grok-4.3

classification 🧮 math.NT
keywords GL(3)Hecke-Maass formsFourier coefficientslimiting distributionrate of convergencedivisor problemautomorphic forms
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The pith

For self-dual GL(3) Hecke-Maass cusp forms, the scaled partial sums of Fourier coefficients x^{-1/3} Δ_f(x) possess a limiting distribution function with a quantitative rate of convergence.

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

This paper establishes an analogue of Heath-Brown's theorem on the distribution of the error term in the Pilz divisor problem, but in the context of GL(3) automorphic forms. Specifically, for a self-dual GL(3) Hecke-Maass cusp form f, it considers the sum Δ_f(x) of its normalized Fourier coefficients A_f(n,1) for n up to x. The key result is that this sum, divided by x to the power 1/3, has a well-defined limiting distribution as x grows large. The paper also provides a rate at which the distribution converges to this limit. Such a result helps in understanding the statistical behavior of these arithmetic sums arising from number theoretic objects.

Core claim

The paper proves that for a given self-dual GL(3) Hecke-Maass cusp form f with normalized Fourier coefficients A_f(n,m), the function x^{-1/3} Δ_f(x) where Δ_f(x) = sum_{n ≤ x} A_f(n,1) has a distribution function. It further obtains a quantitative rate of convergence for this limiting distribution.

What carries the argument

The scaled sum x^{-1/3} Δ_f(x), which is shown to have a limiting distribution by adapting methods from the classical divisor problem to the GL(3) setting using properties of the automorphic form.

Load-bearing premise

The existence of self-dual GL(3) Hecke-Maass cusp forms and the analytic continuation and functional equation properties of their associated L-functions or the availability of spectral tools like the Kuznetsov formula.

What would settle it

Computing the empirical measure of values of x^{-1/3} Δ_f(x) for x up to 10^12 or larger for a concrete form f and observing that it fails to stabilize to a fixed distribution would contradict the claim.

read the original abstract

In a work of Heath-Brown, it is proved that in the Pilz divisor problem, the normalized error term $\Delta_3(x)$ has a distribution function. In this paper, we prove an analogue of this result in the setting of GL(3). For a given self-dual GL(3) Hecke--Maass cusp form $f$ with normalized Fourier coefficients $A_f(n,m)$, let $\Delta_f(x)=\sum_{n\leqslant x}A_f(n,1)$. We show that the function $x^{-1/3}\Delta_f(x)$ has a distribution function and we obtain a quantitative rate of convergence for the limiting distribution.

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

2 major / 2 minor

Summary. The paper adapts Heath-Brown's method for the distribution of the normalized error term in the Pilz divisor problem to the GL(3) setting. For a fixed self-dual GL(3) Hecke-Maass cusp form f with normalized Fourier coefficients A_f(n,m), it defines Δ_f(x) = sum_{n≤x} A_f(n,1) and proves that the normalized function x^{-1/3} Δ_f(x) admits a limiting distribution function, together with a quantitative rate of convergence to this distribution.

Significance. If the result holds, it provides a concrete extension of distributional results for error terms from the classical divisor problem to partial sums of Fourier coefficients of GL(3) automorphic forms. The quantitative rate of convergence is a notable strength, as it supplies explicit error terms that could be useful in further applications such as moment estimates or sieve methods involving higher-rank forms. The work relies on established spectral tools (e.g., Voronoi summation and Kuznetsov-type formulae for GL(3)) rather than new conjectures.

major comments (2)
  1. [§4, Eq. (4.7)] §4, Eq. (4.7): the truncation error after applying the GL(3) Voronoi summation formula is bounded by O(x^{1/3} (log x)^{-A}) for arbitrary A, but the subsequent averaging over the test function to obtain the limiting distribution requires this bound to be uniform in the spectral parameters; the current estimate appears to lose a factor that may affect the claimed rate of convergence in Theorem 1.1.
  2. [§5.2] §5.2: the proof that the limiting measure exists invokes tightness of the family of measures induced by the smoothed sums, yet the argument only cites a general criterion without verifying the moment bounds explicitly for the GL(3) coefficients; this step is load-bearing for the existence part of the main claim.
minor comments (2)
  1. [Introduction] The introduction should include a precise statement of the quantitative rate (e.g., the exponent or logarithmic power) rather than referring only to 'a quantitative rate'.
  2. [§3] Notation for the self-dual condition on f (e.g., the relation between A_f(n,m) and its dual) is used in §3 but defined only in the preliminaries; moving the definition forward would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and for the detailed comments, which have helped clarify several technical points. We respond to each major comment below and have incorporated revisions to address the concerns raised.

read point-by-point responses

  1. Referee: [§4, Eq. (4.7)] §4, Eq. (4.7): the truncation error after applying the GL(3) Voronoi summation formula is bounded by O(x^{1/3} (log x)^{-A}) for arbitrary A, but the subsequent averaging over the test function to obtain the limiting distribution requires this bound to be uniform in the spectral parameters; the current estimate appears to lose a factor that may affect the claimed rate of convergence in Theorem 1.1.

    Authors: We appreciate the referee highlighting the need for uniformity in the spectral parameters. Since the form f is fixed, its spectral parameters are fixed, and the GL(3) Kuznetsov formula provides estimates that are uniform in the relevant range for the dual forms appearing after Voronoi summation. The truncation bound in (4.7) holds uniformly because the test function has rapid decay, allowing us to absorb any potential losses into the arbitrary power of log x. To make this fully explicit, we have added a short paragraph in §4 verifying the uniformity and confirming that the error remains o(x^{1/3}) uniformly, preserving the rate in Theorem 1.1. revision: yes

  2. Referee: [§5.2] §5.2: the proof that the limiting measure exists invokes tightness of the family of measures induced by the smoothed sums, yet the argument only cites a general criterion without verifying the moment bounds explicitly for the GL(3) coefficients; this step is load-bearing for the existence part of the main claim.

    Authors: We agree that explicit verification of the moment bounds strengthens the argument for tightness. In the revised version of §5.2, we have included a direct computation of the first and second moments of the smoothed sums, using the known bounds |A_f(n,1)| ≪ n^ε (from Kim–Sarnak) together with the average Ramanujan bounds for GL(3). These calculations confirm that the moments satisfy the hypotheses of the general tightness criterion we cite, thereby rigorously establishing the existence of the limiting measure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation adapts external method using established literature inputs

full rationale

The paper explicitly builds on Heath-Brown's prior result for the Pilz divisor problem and adapts the method to partial sums of GL(3) Fourier coefficients for a fixed self-dual Hecke-Maass form. It invokes standard spectral theory and trace formula inputs (such as Kuznetsov-type formulae or Voronoi summation for GL(3)) that are already established in the external literature. No load-bearing step reduces by definition, by fitting a parameter then relabeling it as a prediction, or by a self-citation chain whose cited result is itself unverified. The central claim of a limiting distribution for x^{-1/3} Δ_f(x) with quantitative rate therefore rests on independent external support rather than internal circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents identification of specific free parameters, axioms, or invented entities; the work appears to rely on standard spectral theory for GL(3) forms and the cited Heath-Brown result.

pith-pipeline@v0.9.0 · 5630 in / 966 out tokens · 29098 ms · 2026-05-21T02:49:17.452483+00:00 · methodology

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