Fourier optimization and quadratic forms

TL;DR

This paper applies Fourier analysis to quadratic forms to improve error estimates in counting integer representations, leading to new results on prime distributions represented by these forms.

Contribution

It introduces Fourier analysis techniques to enhance error bounds in quadratic form representations and derives new prime distribution results related to these forms.

Findings

Improved error terms in counting representations of integers by quadratic forms.

Unconditional Brun-Titchmarsh-type bounds in short intervals for primes represented by quadratic forms.

Conditional Cramér-type bounds on gaps between primes represented by quadratic forms.

Abstract

We prove several results about integers represented by positive definite quadratic forms, using a Fourier analysis approach. In particular, for an integer $\ell\geq 1$, we improve the error term in the partial sums of the number of representations of integers that are a multiple of $\ell$. This allows us to obtain unconditional Brun-Titchmarsh-type results in short intervals, and a conditional Cram\'er-type result on the maximum gap between primes represented by a given positive definite quadratic form.