Asymptotics for the Dirichlet coefficients of symmetric power $L$-functions
Volume 199 / 2021
Acta Arithmetica 199 (2021), 253-268
MSC: Primary 11F30; Secondary 11F66.
DOI: 10.4064/aa191112-24-12
Published online: 14 June 2021
Abstract
Let $L(\mathop {\rm sym}^jf, s)$ be the $j$th symmetric power $L$-function attached to a holomorphic Hecke eigencuspform $f(z)$ for the full modular group $\Gamma =\mathrm {SL}(2,\mathbb {Z})$, and $\lambda _{\mathop {\rm sym}^jf}(n)$ denote its $n$th Dirichlet coefficient. We establish asymptotic formulas for $\sum _{n\leq x}\lambda _{\mathop {\rm sym}^2f}^j(n)$ and $\sum _{n\leq x}\lambda _{\mathop {\rm sym}^jf}^2(n)$ for $j=3,4,5$, $6,7,8,$ and obtain two non-trivial upper bounds for the mean-square of the error term related to $\sum _{n\leq x}\lambda ^2_{\mathop {\rm sym}^jf}(n)$ for $j=7,8.$
