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Higher power sums of Fourier coefficients of symmetric square L-functions

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  21 October 2025

Jinzhi Feng Open the ORCID record for Jinzhi Feng [Opens in a new window]
Jinzhi Feng*
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong, China
*
Email: fengjinzhi@mail.sdu.edu.cn
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Abstract

Let f(z) be the normalized primitive holomorphic Hecke eigenforms of even integral weight k for the full modular group $SL(2,\mathbb{Z})$ and denote $L(s,\mathrm{sym}^{2}f)$ be the symmetric square L-function attached to f(z). Suppose that $\lambda_{\mathrm{sym}^{2}f}(n)$ be the $\mathrm{Fourier}$ coefficient of $L(s,\mathrm{sym}^{2}f)$. In this paper, we investigate the sum $\sum\limits_{n\leqslant x}\lambda^{j}_{\mathrm{sym}^{2}f }(n) $ for $j\geqslant 3$ and obtain some new results which improve on previous error estimates. We also consider the sum $\sum\limits_{n\leqslant x}\lambda^{j}_{f }(n^{2})$ and get some similar results.

Keywords

cusp formFourier coefficientsymmetric square L-functionvariance

MSC classification

Primary: 11F30: Fourier coefficients of automorphic forms
Secondary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations

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Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 22
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Edinburgh Mathematical Society

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