arxiv: 2605.06363 · v1 · submitted 2026-05-07 · 🧮 math.NT

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On algebraic twists with composite moduli, II

Philippe Michel, Yongxiao Lin

Pith reviewed 2026-05-08 05:25 UTC · model grok-4.3

classification 🧮 math.NT

keywords correlation sumsGL(3) automorphic formstrace functionscomposite modulialgebraic twistsanalytic number theoryautomorphic coefficients

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The pith

The bounds on correlation sums of GL_3 automorphic coefficients with trace functions extend from prime moduli to composite moduli.

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

The authors examine sums that pair the Fourier coefficients of automorphic forms on GL_3 over the rationals with algebraic trace functions, now allowing the modulus to be any composite integer. This continues earlier work restricted to prime moduli and shows that the same quality of bounds remains available. The extension removes a common technical barrier because many applications in analytic number theory naturally involve sums over all integers rather than only primes. A sympathetic reader would therefore expect the new results to plug directly into existing estimates without further adjustment.

Core claim

We establish square-root cancellation bounds for the correlation sums of GL_3 automorphic coefficients with trace functions modulo composite q, by verifying that the analytic properties of the associated Dirichlet series and L-functions carry over unchanged from the prime-modulus setting treated in prior work.

What carries the argument

The correlation sum pairing GL_3 automorphic coefficients with trace functions modulo composite q, bounded via analytic continuation of the corresponding L-functions.

If this is right

The bounds now apply uniformly when averaging over all moduli up to a given size rather than only primes.
Error terms in applications to character sums or exponential sums become independent of the prime-versus-composite distinction.
Subconvexity or equidistribution results that previously required prime moduli can be stated for general moduli.

Where Pith is reading between the lines

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

The same extension technique may apply when the modulus has an arbitrary number of prime factors, provided uniformity in the estimates is preserved.
Numerical checks for small composite moduli would provide direct evidence that the analytic continuation step works as asserted.
The results suggest that trace-function methods over the integers can be formulated without restricting to prime-power or prime moduli.

Load-bearing premise

The analytic properties and bounds established in the authors' prior works extend without essential change to the composite-modulus setting.

What would settle it

An explicit computation of the correlation sum for a fixed GL_3 form, a concrete trace function, and a small composite modulus such as q=4 or q=6 that produces a value exceeding the claimed bound by a large factor.

read the original abstract

We study bounds for correlation sums of automorphic coefficients on $\mathrm{GL}_{3,\mathbb{Q}}$ with trace functions of composite moduli. This is a sequel to our previous works with E. Kowalski and W. Sawin.

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

Lin and Michel extend the GL(3) correlation sum bounds to composite moduli by CRT decomposition and uniform control on L-functions and conductors from the prior papers.

read the letter

Lin and Michel have worked out bounds for the correlation sums of GL(3) automorphic coefficients against trace functions when the modulus is composite. The key step is using the Chinese Remainder Theorem to reduce to prime power moduli and then applying the estimates from their previous papers uniformly. This extension is done cleanly. The manuscript supplies the adaptations needed for the composite case, including control over the conductors of the associated L-functions. That addresses the main concern about whether the prior analytic properties carry over without essential change. The soft spots are limited. As a sequel, it does not introduce a new framework, so its value is in the verification that the composite setting works at the same level. If there is any degradation in the saving or in the range of moduli allowed, the paper should make that clear for users. Readers who are following this line of work on trace functions and automorphic forms will find it directly applicable. It is narrow in scope but fills a practical gap for anyone needing to work with composite moduli in these sums. The paper shows clear thinking in how it builds on the literature. It deserves a serious referee, and I recommend sending it out for peer review.

Referee Report

0 major / 2 minor

Summary. The paper establishes bounds for correlation sums of GL(3,Q) automorphic coefficients against trace functions modulo composite integers. It is a sequel to the authors' prior works with Kowalski and Sawin, extending the prime-modulus results via CRT decomposition of the modulus together with uniform control on the associated L-functions and conductors.

Significance. The extension supplies a necessary technical step for applying these correlation bounds in settings where the modulus is composite, which covers most arithmetic applications. The manuscript explicitly addresses the reduction to the prime-power case and the required uniformity, so the central claim rests on verifiable adaptations rather than unstated assumptions.

minor comments (2)

The abstract states the main object of study but does not record the precise form of the bound or the dependence on the composite modulus; adding one sentence would make the result immediately usable for readers.
Notation for the composite modulus q = q1 q2 and the associated trace function is introduced in the introduction; a short table or displayed list of the main symbols and their ranges would improve readability in later sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript. We are pleased that the referee recognizes the paper as a necessary technical extension of our prior work with Kowalski and Sawin, achieved through CRT decomposition together with uniform control on L-functions and conductors, and that this supplies a verifiable step toward composite-modulus applications.

Circularity Check

0 steps flagged

No significant circularity: self-contained extension with explicit adaptations

full rationale

The paper is explicitly a sequel extending prior results on correlation sums for GL(3) automorphic coefficients against trace functions, now to composite moduli. It addresses the key assumption by supplying adaptations via CRT decomposition of the modulus and uniform control on the associated L-functions and conductors. No load-bearing derivation step reduces by the paper's own equations or citations to a self-definition, a fitted input renamed as prediction, or an unverified self-citation chain; the central claims remain independent of the inputs once the stated modifications are applied.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract only; the central claim rests on the analytic continuation and growth properties of GL(3) automorphic forms and the Weil-Deligne bound or similar estimates for trace functions, all taken from prior literature.

axioms (2)

domain assumption Standard analytic properties of automorphic forms on GL(3) over Q, including functional equations and convexity bounds.
Invoked implicitly when discussing correlation sums of automorphic coefficients.
domain assumption Trace functions satisfy the Riemann hypothesis over finite fields and have bounded conductor.
Required to control the sums when the modulus is composite.

pith-pipeline@v0.9.0 · 5310 in / 1201 out tokens · 36894 ms · 2026-05-08T05:25:00.634574+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 1 canonical work pages

[1]

Aggarwal,A new subconvex bound forGL(3)L-functions in thet-aspect, Int

[Agg21] K. Aggarwal,A new subconvex bound forGL(3)L-functions in thet-aspect, Int. J. Number Theory17 (2021), no. 5, 1111–1138. [DFI93] W. Duke, J. Friedlander, and H. Iwaniec,Bounds for automorphicL-functions, Invent. Math.112(1993), no. 1, 1–8. [FKM15] ´E. Fouvry, E. Kowalski, and Ph. Michel,Algebraic twists of modular forms and Hecke orbits, Geom. Func...

work page arXiv 2021
[2]

Kowalski, Ph

[KMS17] E. Kowalski, Ph. Michel, and W. Sawin,Bilinear forms with Kloosterman sums and applications, Ann. of Math. (2)186(2017), no. 2, 413–500. [LM24] Y. Lin and Ph. Michel,On algebraic twists with composite moduli, Ramanujan J.63(2024), no. 3, 803–837. [LMS23] Y. Lin, Ph. Michel, and W. Sawin,Algebraic twists ofGL 3 ×GL 2 L-functions, Amer. J. Math.145(...

2017