Sums of algebraic trace functions twisted by arithmetic functions

TL;DR

This paper establishes new bounds for short sums of algebraic trace functions twisted by arithmetic functions like Mobius and divisor functions, improving on previous length thresholds for nontrivial estimates.

Contribution

It provides novel bounds for sums of trace functions twisted by arithmetic functions, extending the range of short intervals where nontrivial estimates hold.

Findings

Bounds are nontrivial for intervals of length at least p^{1/2+ε}.

Improves previous bounds for sums twisted by Mobius and divisor functions.

Includes classical objects like Kloosterman sums within the new bounds.

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 Mobius 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 at least $p^{1/2+\varepsilon}$ with an arbitrary fixed $\varepsilon >0$, which is shorter than the length at least $p^{3/4+\varepsilon}$ in the case of the Mobius function and at least $p^{2/3+\varepsilon}$ in the case of the divisor function required in recent results of {\'E}.~Fouvry, E.~Kowalski and P.~Michel (2014) and E.~Kowalski, P. ~Michel and W.~Sawin (2018), respectively.