On the average behavior of the Fourier coefficients of $j^{th}$ symmetric power $L$-function over a certain sequences of positive integers

Abstract

In this paper, we investigate the average behavior of the $n^{th}$ normalized Fourier coefficients of the $j^{th}$ ($j\ge 2$ be any fixed integer) symmetric power $L$-function (i.e., $L(s,sy{m}^{j}f)$), attached to a primitive holomorphic cusp form $f$ of weight $k$ for the full modular group $SL(2,\mathbb{Z})$ over a certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum $$\sum _{\stackrel{a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2\le x}{(a_1,a_2,a_3,a_4,a_5,a_6)\in {\mathbb{Z}}^6}}{\lambda }_{sy{m}^{j}f}^2(a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2),$$ where $x$ is sufficiently large, and $$L(s,sy{m}^{j}f):=\sum _{n=1}^{\infty }\frac{{\lambda }_{sy{m}^{j}f}(n)}{n^s}.$$ When $j=2$, the error term which we obtain, improves the earlier known result.