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Article

Hua, Guodong
On the higher power moments of cusp form coefficients over sums of two squares. (English). Czechoslovak Mathematical Journal, vol. 72 (2022), issue 4, pp. 1089-1104
MSC: 11F11, 11F30, 11F66 | MR 4517598 | Zbl 07655785 | DOI: 10.21136/CMJ.2022.0358-21
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Keywords:
Fourier coefficient; automorphic $L$-function, Langlands program
Summary:
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.
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