Czechoslovak Mathematical Journal, first online, pp. 1-14
A note on average behaviour of the Fourier coefficients of $j$th symmetric power $L$-function over certain sparse sequence of positive integers
Youjun Wang
Received January 18, 2024. Published online April 29, 2024.
Abstract: Let $j\geq2$ be a given integer. Let $H_k^*$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\geq2$ for the full modulo group ${\rm SL}(2,\mathbb{Z})$. For $f\in H_k^*$, denote by $\lambda_{{\rm sym}^jf}(n)$ the $n$th normalized Fourier coefficient of $j$th symmetric power $L$-function ($L(s, {\rm sym}^jf)$) attached to $f$. We are interested in the average behaviour of the sum $\sum_{n=a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2\leq x \atop(a_1,a_2,a_3,a_4,a_5,a_6)\in\mathbb{Z}^{ 6}} \lambda_{{\rm sym}^jf}^2(n)$, where $x$ is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).
Keywords: cusp form; Fourier coefficient; symmetric power $L$-function
Affiliations: Youjun Wang, School of Mathematics and Statistics, Henan University, 1 Jinming Road, Kaifeng 475004, P. R. China, e-mail: math_wyj@henu.edu.cn
