Divisor problems for restricted Fourier coefficients of modular forms

TL;DR

This paper studies the average behavior of the divisor function applied to Fourier coefficients of modular forms, focusing on primes with specific angle constraints.

Contribution

It introduces new methods to analyze divisor problems for restricted Fourier coefficients of modular forms, expanding understanding of their average properties.

Findings

Derived bounds for the average of divisor functions of Fourier coefficients

Analyzed the effect of angle constraints on prime Fourier coefficients

Extended divisor problem techniques to modular form coefficients

Abstract

Let $d(n)$ be the number of divisors of $n$. We investigate the average value of $d(a_f(p))^r$ for $r$ a positive integer and $a_f(p)$ the $p$-th Fourier coefficient of a cuspidal eigenform $f$ having integral Fourier coefficients, where $p$ is a prime subject to a constraint on the angle associated with the normalized Fourier coefficient.