Discrete mean square of the coefficients of symmetric square L-functions on certain sequence of positive numbers

In this paper, we will be concerned with the average behavior of the nth\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\mathrm{th}$$\end{document} normalized Fourier coefficients of symmetric square L-function (i.e., L(s,sym2f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(s,sym^{2}f)$$\end{document}) over certain sequence of positive integers. Precisely, we prove an asymptotic formula for ∑a2+b2+c2+d2≤x(a,b,c,d)∈Z4λsym2f2(a2+b2+c2+d2),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathop {\sum }\limits _{\begin{array}{c} a^{2}+b^{2}+c^{2}+d^{2}\le {x} \\ (a,b,c,d)\in {\mathbb {Z}}^{4} \end{array}}\uplambda ^{2}_{sym^{2}f}(a^{2}+b^{2}+c^{2}+d^{2}), \end{aligned}$$\end{document}where x≥x0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\ge {x_{0}}$$\end{document} (sufficiently large), and L(s,sym2f):=∑n=1∞λsym2f(n)ns.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} L(s,sym^{2}f):= \mathop {\sum }\limits _{n=1}^{\infty }\dfrac{\uplambda _{sym^{2}f}(n)}{n^{s}}. \end{aligned}$$\end{document}