Estimates for representation numbers of quadratic forms

Abstract

Let $f$ be a primitive positive integral binary quadratic form of discriminant $-D$, and let $r_f(n)$ be the number of representations of $n$ by $f$ up to automorphisms of $f$. In this article, we give estimates and asymptotics for the quantity $\sum _{n\le x}{r}_f(n){}^{\beta }$ for all $\beta \ge 0$ and uniformly in $D=o(x)$. As a consequence, we get more-precise estimates for the number of integers which can be written as the sum of two powerful numbers