A Version of the Circle Method for the Representation of Integers by Quadratic Forms

TL;DR

This paper adapts the circle method with Gaussian weights to derive an asymptotic formula for how often non-zero integers are represented by non-singular quadratic forms in four or more variables.

Contribution

It introduces a modified circle method approach with Gaussian weights to improve the analysis of representations by quadratic forms.

Findings

Derived an asymptotic formula for the density of integer representations.

Extended the circle method to quadratic forms with Gaussian weights.

Applicable to non-singular quadratic forms in at least four variables.

Abstract

We apply the circle method with a Gaussian weight to obtain an asymptotic formula for the density of representations of non-zero integers by non-singular quadratic forms in at least four variables.