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The Average Behavior of Fourier Coefficients of Symmetric Power L-Functions

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Abstract

Let \(H_k^*\) be the set of all normalized primitive holomorphic cusp forms of even weight \(k\ge 2\) for \(SL_2(\mathbb {Z})\). For \(f\in H_k^*\), let \(\lambda _{\textrm{sym}^jf}(n)\) be the nth normalized Fourier coefficient of the jth symmetric power L-function attached to f. In this paper, we establish asymptotic formulas for the power sums of \(\lambda _{\textrm{sym}^jf}(n)\) and improve previous results.

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Acknowledgements

The author would like to thank the referees for many valuable comments on the manuscript. This work is supported by the National Natural Science Foundation of China (Grant No. 12171286).

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  1. School of Mathematics and Statistics, Shandong Normal University, Ji’nan, 250358, Shandong, China

    Huafeng Liu

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  1. Huafeng Liu
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Communicated by Rosihan M. Ali.

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Liu, H. The Average Behavior of Fourier Coefficients of Symmetric Power L-Functions. Bull. Malays. Math. Sci. Soc. 46, 193 (2023). https://doi.org/10.1007/s40840-023-01586-z

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