Divisor moments of polynomials in Fourier coefficients of modular forms

Abstract

We study higher moments of the divisor function evaluated at polynomial expressions in the Fourier coefficients of a non-CM newform. The logarithmic exponent appearing in our estimates depends only on the number of irreducible factors of the polynomial and remains unchanged under a Sato--Tate restriction. The proof combines an effective Chebotarev theorem, or an effective Chebotarev--Sato--Tate theorem, with the arithmetic of joint cycle types and a mean value estimation for multivariable multiplicative functions with Frobenian support.

Disclosure

“ons and valuable suggestions for this work. The author is also grateful to M. Ram Murty for suggesting that he revisit Erdös’s paper [Erd52], which helped remove some assumptions imposed in an earlier version of the paper. The author used OpenAI’s ChatGPT (GPT-5.6 Thinking, accessed in July 2026) to assist with language editing, the organization of parts of the exposition, and preliminary consistency checks. All mathematical statements and arguments were independently established, reviewed,”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T18:59:59.852120Z

Structural counts