Fourier optimization and 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 ℓ≥1⁠, we improve the error term in the partial sums of the number of representations of integers that are a multiple of ℓ⁠. This allows us to obtain unconditional Brun–Titchmarsh-type results in short intervals and a conditional Cramér-type result on the maximum gap between primes represented by a given positive definite quadratic form.