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On higher moments of Dirichlet coefficients attached to symmetric square L-functions over certain sparse sequence

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Abstract

Let \(2\le j\le 8\) be any fixed positive integer. Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group \(\Gamma =SL(2,\mathbb {Z})\). Denote by \(\lambda _{\text {sym}^{2}f}(n)\) the nth normalized coefficient of the Dirichlet expansion of the symmetric square L-function \(L(\text {sym}^{2}f,s)\) attached to f. In this paper, we are interested in the average behaviour of the following summatory function

$$\begin{aligned} \sum _{\begin{array}{c} a^{2} + b^{2} + c^{2} + d^{2}\le x \\ (a,b,c,d)\in \mathbb {Z}^{4} \end{array}} \lambda _{\text {sym}^{2}f}^{j}(a^{2}+b^{2}+c^{2}+d^{2}) \end{aligned}$$

for \(x\ge x_{0}\)(sufficiently large), which improves and generalizes the recent works of Sharma and Sankaranarayanan (Res Number Theory 8:19, 2022, Rend Circ Mat Palermo II Ser 72:1399–1416, 2023).

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Acknowledgements

The first author would like to extend his sincere gratitude to Professors Guangshi Lü, Bingrong Huang and Dr. Zhiwei Wang for their constant encouragement and valuable suggestions. The authors are extremely grateful to the anonymous referees for their meticulous checking, for thoroughly reporting countless typos and inaccuracies as well as for their valuable comments. These corrections and additions have made the manuscript clearer and more readable.

Funding

This work was supported in part by The National Key Research and Development Program of China (Grant No. 2021YFA1000700) and Natural Science Basic Research Program of Shaanxi (Program Nos. 2023-JC-QN-0024, 2023-JC-YB-077).

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  1. School of Mathematics and Statistics, Weinan Normal University, Weinan, 714099, Shaanxi, China

    Guodong Hua, Bin Chen, Lijing Pan & Xiaofang Chen

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  1. Guodong Hua
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Hua, G., Chen, B., Pan, L. et al. On higher moments of Dirichlet coefficients attached to symmetric square L-functions over certain sparse sequence. Rend. Circ. Mat. Palermo, II. Ser 72, 4195–4208 (2023). https://doi.org/10.1007/s12215-023-00898-0

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