On Representation of Integers by Binary Quadratic Forms

Given a negative $D>-(\log\ X)^{\log\ 2-\delta}$, 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 $a\in\mathbb Z$ is fixed. Combined with the $\frac {1}{2}$-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 Bourgain and Fuchs in [3], is a lower bound on the number of primes which come up as curvatures in a given primitive integer Apollonian circle packing.