On sums of fourier coefficients of automorphic forms for $gl_r$

Jaban Meher

Published 2014-12-30Version 1

For $r\ge 2$, let $\pi$ be an irreducible cuspidal automorphic representation of $GL_r(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. Let $a_\pi(n)$ be the $n^{th}$ coefficients of the $L$-function attached to $\pi$. Goldfeld and Sengupta have recently obtained a bound for $\sum_{n\le x} a_\pi(n)$ as $x \rightarrow \infty$. For $r\ge 3$ and $\pi$ not a symmetric power of $GL_r(\mathbb{A}_{\mathbb{Q}})$ cuspidal automorphic representation, their bound is better than all previous bounds. The goal of this paper is to further improve the bound of Goldfeld and Sengupta.