On the asymptotics of coefficients of Rankin–Selberg L-functions

Let f and g be two different holomorphic cusp froms or Maass cusp forms 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}. We are interested in coefficients of Rankin–Selberg L-functions, and establish some bounds for ∑n≤xλsymif×symjg(n),∑n≤xλf(ni)λg(nj),∑n≤x|λsymif×symjg(n)|,∑n≤x|λf(ni)λg(nj)|,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\sum_{n\leq x} \lambda_{{\rm sym}^if\times {\rm sym}^jg}(n),\quad \sum_{n\leq x}\lambda_f(n^i)\lambda_g(n^j), \\ \sum_{n\leq x} |\lambda_{{\rm sym}^if\times {\rm sym}^jg}(n)|, \quad \sum_{n\leq x}|\lambda_f(n^i)\lambda_g(n^j)|, \end{aligned}$$\end{document} and ∑n≤xmax{|λsymif×symjg(n)|2φ,|λsymif×symjg(n+h)|2φ},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n\leq x} \max \bigl\{|\lambda_{{\rm sym}^if\times {\rm sym}^jg}(n)|^{2\varphi}, |\lambda_{{\rm sym}^if\times {\rm sym}^jg}(n+h)|^{2\varphi} \bigr\}, $$\end{document} where φ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi>0$$\end{document} and h is a fixed positive integer.