On representation of integers by binary quadratic forms

TL;DR

This paper establishes new upper bounds on the count of square-free integers represented by certain binary quadratic forms, and applies these results to prime curvatures in Apollonian circle packings.

Contribution

It introduces novel bounds for integers represented by quadratic forms with growing discriminant and connects these bounds to prime curvatures in circle packings.

Findings

New upper bounds for square-free integers represented by quadratic forms.

Application of bounds to prime curvatures in Apollonian circle packings.

Use of sieve methods to derive lower bounds on represented integers.

Abstract

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 $f$ 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 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 [BF], is a lower bound on the number of primes which come up as curvatures in a given primitive integer Apollonian circle packing.