SUMS OF KLOOSTERMAN SUMS OVER PRIMES IN AN ARITHMETIC PROGRESSION

Alexander Dunn, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review

Abstract

For q prime, X≥1 and coprime u,v∈ℕ, we estimate the sums (Equation Presnted) where Kl 2 (p;q) denotes a normalized Kloosterman sum with modulus q. This is a sparse analog of a recent theorem due to Blomer, Fouvry, Kowalski, Michel and Milićević showing cancellation amongst sums of Kloosterman sums over primes in short intervals. We use an optimization argument inspired by Fouvry, Kowalski and Michel. Our argument compares three different bounds for bilinear forms involving Kloosterman sums. The first input in this method is a bilinear bound we prove using uniform asymptotics for oscillatory integrals due to Kiral, Petrow and Young. In contrast with the case when the sum runs over all primes, we exploit cancellation over a sum of stationary phase integrals that result from a Voronoi type summation. The second and third inputs are deep bilinear bounds for Kloosterman sums due to Fouvry-Kowalski-Michel and Kowalski-Michel-Sawin.

Original language English (US) 319-342 24 Quarterly Journal of Mathematics 70 1 https://doi.org/10.1093/qmath/hay035 Published - Mar 1 2019

ASJC Scopus subject areas

• Mathematics(all)