Czechoslovak Mathematical Journal, Vol. 73, No. 3, pp. 885-901, 2023
On the average behavior of the Fourier coefficients of $j$th symmetric power $L$-function over certain sequences of positive integers
Anubhav Sharma, Ayyadurai Sankaranarayanan
Received August 16, 2022. Published online April 27, 2023.
Abstract: We investigate the average behavior of the $n$th normalized Fourier coefficients of the $j$th ($j \geq2$ be any fixed integer) symmetric power $L$-function (i.e., $L(s,{\rm sym}^jf)$), attached to a primitive holomorphic cusp form $f$ of weight $k$ for the full modular group $SL(2,\mathbb{Z})$ over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum $S_j^*:= \sum_{a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2\leq x \atop (a_1,a_2,a_3,a_4,a_5,a_6)\in\mathbb{Z}^6} \lambda^2_{{\rm sym}^jf}(a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2),$
where $x$ is sufficiently large, and $L(s,{\rm sym}^jf):=\sum_{n=1}^{\infty}\frac{\lambda_{{\rm sym}^jf}(n)}{n^s}.$
When $j=2$, the error term which we obtain improves the earlier known result.
Keywords: nonprincipal Dirichlet character; Hölder's inequality; $j$th symmetric power $L$-function; holomorphic cusp form
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Affiliations: Anubhav Sharma (corresponding author), Ayyadurai Sankaranarayanan, School of Mathematics and Statistics, University of Hyderabad Central University, P.O., Prof. C. R. Rao Road, Gachibowli, Hyderabad-500046, India, e-mail: 19mmpp02@uohyd.ac.in, sank@uohyd.ac.in
