On sums involving coefficients of automorphic $L$-functions

Abstract:

Let $L(s,\pi )$ be the automorphic $L$-function associated to an automorphic irreducible cuspidal representation $\pi$ of $\text {GL}_m$ over $\mathbb {Q}$, and let $a_{\pi }(n)$ be the $n$th coefficient in its Dirichlet series expansion. In this paper we prove that if at every finite place $p$, $\pi _p$ is unramified, then for any $\varepsilon >0$, \begin{equation*} A_{\pi }(x) = \sum _{n \leq x}a_{\pi }(n) \ll _{\varepsilon ,\pi } \begin {cases} x^{\frac {71}{192}+\varepsilon } & \text {if $m=2$},\\ x^{\frac {m^2-m}{m^2+1}+\varepsilon } & \text {if $m \geq 3$}. \end{cases} \end{equation*}