On the higher power moments of cusp form coefficients over sums of two squares

Hua, Guodong

Czechoslovak Mathematical Journal, Tome 72 (2022) no. 4, pp. 1089-1104.

Résumé

Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma ={\rm SL} (2,\mathbb {Z})$. Denote by $\lambda _{f}(n)$ the $n$th normalized Fourier coefficient of $f$. We are interested in the average behaviour of the sum $$ \sum _{a^{2} + b^{2}\leq x} \lambda _{f}^{j}(a^{2}+b^{2}) $$ for $x\geq 1$, where $a,b\in \mathbb {Z}$ and $j\geq 9$ is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power $L$-functions and Rankin-Selberg $L$-functions.

Détail

DOI : 10.21136/CMJ.2022.0358-21

Classification : 11F11, 11F30, 11F66
Keywords: Fourier coefficient; automorphic $L$-function, Langlands program