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 p12+𝜀 with an arbitrary fixed 𝜀 > 0, which is shorter than the required length of at least p34+𝜀 in the case of the Möbius function and of at least p23+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