Averages of shifted convolutions of general divisor sums involving Hecke eigenvalues

Suppose ∑n=1∞a(n)n-s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n=1}^{\infty }a(n) n^{-s}$$\end{document} be a Dirichlet series in the Selberg class of degree d and let E(x) be the arithmetical error term of ∑n⩽xa(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n\leqslant x}a(n)$$\end{document}. By the truncated Tong-type formula of E(x), we can get two kinds of the mean square estimates of E(x) in short intervals of Jutila’s type. Using the estimates, we are able to improve some previous results established by Lü and Wang [9].