On the error terms for representation numbers of quadratic forms

Let f be a primitive positive integral binary quadratic form of discriminant −D, and rf (n) the number of representations of n by f up to automorphisms of f. We first improve the error term E(x) of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sum\limits_{n \leqq x} {r_f (n)^\beta } $$\end{document} for any positive integer β. Next, we give an estimate of ∫1T|E(x)|2x−3/2dx when β = 1.