Counting Square Discriminants

TL;DR

This paper develops an analytic approach using double Dirichlet series to count integral binary quadratic forms with fixed discriminant under coefficient size restrictions.

Contribution

It introduces a novel method analyzing a double Dirichlet series related to automorphic forms for counting quadratic forms.

Findings

Derived asymptotic formulas for the number of quadratic forms with bounded coefficients

Established connections between quadratic form counts and automorphic form properties

Extended classical counting problems using advanced analytic techniques

Abstract

Counting integral binary quadratic forms with certain restrictions is a classical problem. In this paper, we count binary quadratic forms of fixed discriminant given restrictions on the size of their coefficients. We accomplish this by investigating the analytic properties of a certain double Dirichlet series, which is a shifted convolution sum of certain classical automorphic forms.