Discrete mean square estimates for coefficients of symmetric power $L$-functions
Volume 190 / 2019
Acta Arithmetica 190 (2019), 193-208
MSC: 11F30, 11F66.
DOI: 10.4064/aa180819-6-10
Published online: 27 June 2019
Abstract
Let $f$ be a primitive holomorphic Hecke eigenform for $\mathrm{SL}(2, \mathbb{Z})$. Let $L(\mathop{\rm sym}\nolimits^j f, s)$ be the $j$th symmetric power $L$-function associated to $f$, and $\lambda_{\mathop{\rm sym}\nolimits^j f}(n)$ its $n$th Fourier coefficient. We prove asymptotic formulas for the sums \begin{equation*} \sum_{n \leq x} | \lambda_{\mathop{\rm sym}\nolimits^3 f}(n)|^2 \quad \text{and} \quad \sum_{n \leq x} | \lambda_{\mathop{\rm sym}\nolimits^4 f}(n)|^2 \end{equation*} with improved error terms for $x\geq x_0$ (large).
