Fourier optimization and quadratic forms

2020/12/14 by Chirre, Andrés, Quesada-Herrera, Emily
#11E16 #11M26 #11N05 #11R29 #11R42 #42B10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2012.07781

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.

Related

• Fourier optimization and consequences of the generalized Riemann hypothesis
• Fourier optimization, the least quadratic non-residue, and the least prime in an arithmetic progression
• Quadratic minima and modular forms
• Integers represented by positive-definite quadratic forms and Petersson inner products
• Quaternary quadratic forms with prime discriminant