Oscillations of coefficients of Dirichlet series attached to automorphic forms

Abstract

For $m\ge 2$, let $\pi$ be an irreducible cuspidal automorphic representation of $G{L}_m({\mathbb{A}}_{\mathbb{Q}})$ 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 $x\to \infty$. For $m\ge 3$ and $\pi$ not a symmetric power of a $G{L}_2({\mathbb{A}}_{\mathbb{Q}})$-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 Golfeld 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.

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