On Representation of Integers by Binary Quadratic Forms

Abstract

Given a negative D > −(logX)log2−δ , we give a new upper bound on the number of square free integers X which are represented by some but not all forms of the genus of a primitive positive definite binary quadratic form of discriminant D. We also give an analogous upper bound for square free integers of the form q+a X where q is prime and a ∈ Z is fixed. Combined with the 1/2-dimensional sieve of Iwaniec, this yields a lower bound on the number of such integers q+a X represented by a binary quadratic form of discriminant D, where D is allowed to grow with X as above. An immediate consequence of this, coming from recent work of the authors in [3], is a lower bound on the number of primes which come up as curvatures in a given primitive integer Apollonian circle packing.