On the exponent of distribution for convolutions of $\mathrm{GL(2)}$ coefficients to smooth moduli

TL;DR

This paper proves an improved distribution exponent for Hecke eigenvalues of holomorphic cusp forms in arithmetic progressions with square-free moduli.

Contribution

It establishes a new exponent of distribution, specifically  + 1/46, for convolutions of  coefficients in the context of smooth moduli.

Findings

Exponent of distribution  + 1/46 for  coefficients in arithmetic progressions.

Distribution result applies to square-free moduli.

Advances understanding of distribution of Hecke eigenvalues.

Abstract

Let $(\lambda_f(n))_{n\geqslant1}$ be the Hecke eigenvalues of a holomorphic cusp form $f$. We prove that the exponent of distribution of $\lambda_f*1$ in arithmetic progressions is as large as $\frac{1}{2}+\frac{1}{46}$ when the modulus $q$ is square-free.