Vol. 15, No. 9, 2021

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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
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 X such that p2 + 1 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.

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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