Fourier optimization and quadratic forms

We prove several results about integers represented by positive definite quadratic forms, using a Fourier analysis approach. In particular, for an integer $\mathrm{\ell }\ge 1$, we improve the error term in the partial sums of the number of representations of integers that are a multiple of $\mathrm{\ell }$. 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.