On the error terms for representation numbers of quadratic forms (Q532078)

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 \(f\). \textit{V. Blomer} and \textit{A. Granville} [Duke Math. J. 135, No. 2, 261--302 (2006; Zbl 1135.11020)] have studied the asymptotic behaviour of \(\sum_{n \leq x} r_f(n)^\beta\) for integers \(\beta \geq 1\). In the present article, the author improves their error term \(E(x)\) and estimates the integral \(\int_1^T | E(x)| ^2 x^{-3/2}\, dx\) for \(\beta = 1\).