Vol. 304, No. 2, 2020

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Sums of algebraic trace functions twisted by arithmetic functions

Maxim Korolev and Igor Shparlinski

Vol. 304 (2020), No. 2, 505–522
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.

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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
Authors
 Maxim Korolev Department of Number Theory Steklov Mathematical Institute Moscow Russia Igor Shparlinski School of Mathematics and Statistics The University of New South Wales Sydney Australia