Sums of algebraic trace functions twisted by arithmetic functions

Maxim Korolev and Igor Shparlinski

Vol. 304 (2020), No. 2, 505–522

DOI: 10.2140/pjm.2020.304.505

Abstract

We obtain new bounds for short sums of isotypic trace functions associated to some sheaf modulo prime $p$ of bounded conductor, twisted by the Möbius function and also by the generalised divisor function. These trace functions include Kloosterman sums and several other classical number theoretic objects. Our bounds are nontrivial for intervals of length of at least $p^{1∕2+𝜀}$ with an arbitrary fixed $𝜀>0$, which is shorter than the required length of at least $p^{3∕4+𝜀}$ in the case of the Möbius function and of at least $p^{2∕3+𝜀}$ in the case of the divisor function required in recent results of É. Fouvry, E. Kowalski and P. Michel (2014) and E. Kowalski, P. Michel and W. Sawin (2018), respectively.

Keywords

Kloosterman sum, trace function, Möbius function, divisor function

Mathematical Subject Classification 2010

Primary: 11L05, 11N25

Secondary: 11T23

Milestones

Received: 1 May 2018

Revised: 9 June 2019

Accepted: 26 August 2019

Published: 12 February 2020