On sums of Fourier coefficients of cusp forms

TL;DR

This paper estimates the sum of Fourier coefficients of cusp forms, specifically analyzing the summatory function of their Hecke series for both non-holomorphic and holomorphic cases, advancing understanding of their asymptotic behavior.

Contribution

It provides new estimates for the summatory function of Fourier coefficients of cusp forms, covering both non-holomorphic and holomorphic cases, which was previously less understood.

Findings

Estimation of the summatory function for non-holomorphic cusp forms

Analysis of the analogous problem for holomorphic cusp forms

Improved bounds on Fourier coefficient sums

Abstract

The summatory function of $t_j(n^2)$ is estimated, where $H_j(s) = \sum_{n=1}^\infty t_j(n)n^{-s}$ is the Hecke series of a non-holomorphic cusp form. The analogous problem of holomorphic cusp forms is also treated.