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 n≤x r f (n) β for all β ≥ 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.

Counting problems in Apollonian packings

“…Therefore one must understand how exactly |S a | depends on the size of a. This question has been addressed by Blomer and Granville in [6], where the authors give lower and/or upper bounds for the number of integers ≤ X represented by a positive definite binary quadratic form which depends almost solely on the size of the form's discriminant as compared to X. In their notation, let U f (X) be the number of integers less than X represented by f , and let D < 0 denote the discriminant of f .…”

“…We now return to bounding Ω as in (4.8). Using the results of Blomer and Granville in [6], we are able to show that…”

On the error terms for representation numbers of quadratic forms

“…We shift the contour to the line s = 1 2 + ε and pick up the pole at s = 1 which gives the main term. This has been computed in [1]. So we can write it as …”

“…V. Blomer and A. Granville [1] obtained a mean value estimate of r f (n) β , which states that there exist constants a k depending on β and f such that…”

“…The Dirichlet series r f (n) β n −s can be analytically continued to the region 1 2 } and has a pole of order K at s = 1.…”