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 $\ell\ge 1$, 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 Cramér-type result on the maximum gap between primes represented by a given positive definite quadratic form.

On the $q$-analogue of the pair correlation conjecture via Fourier optimization

“…However, this did not lead to any improvement over the results obtained with bandlimited functions in A 1 , even after using polynomials of large degrees, and significantly larger than the degree used in [9]. A similar situation occurred in [11], where the aforementioned results in [6] and [10] were further extended to primes represented by quadratic forms. Therein, bandlimited functions also outperform polynomials times gaussians, unless one uses much larger degrees, which might not be feasible.…”

A Word from... Keri Ann Sather-Wagstaff

“…Emily Quesada-Herrera (she/ her/hers) is a Costa Rican trans woman postdoc at Graz University of Technology (Austria) working on the interface of number theory and harmonic analysis. Her recent paper [CQH22] with Chirre investigates the classical problem of representing integers and primes by quadratic forms using tools from analytic and algebraic number theory and Fourier analysis to obtain new estimates in this area. As an application, for primes of the form 𝑥 2 + 27𝑦 2 studied by Euler and Gauss, assuming the generalized Riemann hypothesis, she shows that there is always such a prime in the short interval [𝑥, 𝑥+5.52√𝑥 log 𝑥], for 𝑥 ≫ 0.…”