On the error terms for representation numbers of quadratic forms

Abstract

Let f be a primitive positive integral binary quadratic form of discriminant −D, and r f (n) the number of representations of n by f up to automorphisms of f . We first improve the error term E(x) of n x r f (n) β for any positive integer β. Next, we give an estimate of T 1 E(x) 2 x − 3 2 dx when β = 1.

Fourier optimization and quadratic forms

“…We would like to thank David de Laat for carrying out computations to obtain lower bound for C + (28) and Table 1, via semidefinite programming. We are grateful to Emanuel Carneiro, Micah B. Milinovich, Kristian Seip, Jesse Thorner and Asif Zaman for their helpful comments.…”

“…Higher moments of r f (n) have also been studied by Blomer and Granville [3]. Later, Xu [28] gave some improvements in their error terms. Additionally, he proved that, when ℓ = 1 in Theorem 1, the optimal error term in (1.2) satisfies Ω(D 1/4 x 1/4 ), which generalizes the classical omega result given originally by Hardy and Landau (see [17]).…”