Arithmetic exponent pairs for algebraic trace functions and applications

Wu, Jie; Xi, Ping

doi:10.2140/ant.2021.15.2123

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Arithmetic exponent pairs for algebraic trace functions and applications

Jie Wu and Ping Xi

Appendix: Will Sawin

Vol. 15 (2021), No. 9, 2123–2172

DOI: 10.2140/ant.2021.15.2123

Abstract

We study short sums of algebraic trace functions via the $q$ -analogue of the van der Corput method, and develop theory of arithmetic exponent pairs that coincide with the classical case when the moduli have sufficiently good factorizations. As an application, we prove a quadratic analogue of the Brun–Titchmarsh theorem on average, bounding the number of primes $p \leq X$ such that $p^{2} + 1 \equiv 0 mod q$ . The other two applications include a larger level of distribution of divisor functions in arithmetic progressions and a sub-Weyl subconvex bound of Dirichlet $L$ -functions studied previously by Irving.

Keywords

$q$-analogue of the van der Corput method, arithmetic exponent pairs, trace functions of $\ell$-adic sheaves, Brun–Titchmarsh theorem, linear sieve

Mathematical Subject Classification

Primary: 11L05, 11L07, 11N13, 11N36, 11T23

Secondary: 11M06, 11N37

Milestones

Received: 30 April 2018

Revised: 27 December 2020

Accepted: 15 April 2021

Published: 23 December 2021

Authors

Jie Wu
School of Mathematics and Statistics Qingdao University Qingdao China	CNRS, UMR 8050 Université Paris-Est Créteil Paris France

Ping Xi
School of Mathematics and Statistics Xi’an Jiaotong University Xi’an China

Will Sawin
Department of Mathematics Columbia University New York, NY United States

Vol. 15, No. 9, 2021