Oscillations of coefficients of Dirichlet series attached to automorphic forms

Abstract

For $m\ge 2$, let $\pi$ be an irreducible cuspidal automorphic representation of $GL_m(\mathbb {A}_{\mathbb {Q}})$ with unitary central character. Let $a_\pi (n)$ be the $n^{th}$ coefficient 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 $m\ge 3$ and $\pi$ not a symmetric power of a $GL_2(\mathbb {A}_{\mathbb {Q}})$-cuspidal automorphic representation with not all finite primes unramified for $\pi$, their bound is better than all previous bounds. In this paper, we further improve the bound of Goldfeld and Sengupta. We also prove a quantitative result for the number of sign changes of the coefficients of certain automorphic $L$-functions, provided the coefficients are real numbers.