Estimates for representation numbers of binary quadratic forms and Apollonian circle packings

Abstract

Fix a primitive, positive definite binary quadratic form g with integer coefficients. We prove asymptotic formulas for sums of the form ∑rg(n)β and ∑rg⁎(n)β, where β≥0 and rg(n), resp. rg⁎(n), denote the number of inequivalent representations, resp. proper inequivalent representations, of n by g. These estimates generalize a previous result by Blomer and Granville (2006) by allowing for non-fundamental discriminants and also clarify some details in the proof of the positive density conjecture for integral Apollonian circle packings by Bourgain and Fuchs (2011).