On the average behavior of the Fourier coefficients of jth symmetric power L-function over certain sequences of positive integers

We investigate the average behavior of the nth normalized Fourier coefficients of the jth (j ≽ 2 be any fixed integer) symmetric power L-function (i.e., L(s,symjf)), attached to a primitive holomorphic cusp form f of weight k for the full modular group SL(2,ℤ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SL(2,\mathbb{Z})$$\end{document} over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum Sj∗:=∑a12+a22+a32+a42+a52+a62⩽x(a1,a2,a3,a4,a5,a6)∈ℤ6λsymjf2(a12+a22+a32+a42+a52+a62),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_j^ *: = \sum\limits_{\matrix{{a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 + a_6^2x} \cr {({a_1},{a_2},{a_3},{a_4},{a_5},{a_6}) \in {\mathbb{Z}^6}} \cr}} {\lambda _{{\rm{sy}}{{\rm{m}}^j}f}^2(a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 + a_6^2),} $$\end{document} where x is sufficiently large, and L(s,symjf):=∑n=1∞λsymjf(n)ns.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(s,{\rm{sy}}{{\rm{m}}^j}f): = \sum\limits_{n = 1}^\infty {{{{\lambda _{{\rm{sy}}{{\rm{m}}^j}f}}(n)} \over {{n^s}}}}.$$\end{document} When j = 2, the error term which we obtain improves the earlier known result.