Czechoslovak Mathematical Journal, Vol. 72, No. 4, pp. 1089-1104, 2022


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

Guodong Hua

Received September 26, 2021.   Published online August 8, 2022.

Abstract:  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 $x\geq1$, where $a,b\in\mathbb{Z}$ and $j\geq9$ 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.
Keywords:  Fourier coefficient; automorphic $L$-function, Langlands program
Classification MSC:  11F11, 11F30, 11F66
DOI:  10.21136/CMJ.2022.0358-21


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Affiliations:   Guodong Hua, School of Mathematics and Statistics, Weinan Normal University, Chaoyang Street, Shaanxi, Weinan 714099, P. R. China, e-mail: gdhua@mail.sdu.edu.cn


 
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