Kloosterman sums in residue classes

TL;DR

This paper establishes upper bounds for sums of Kloosterman sums with arithmetic weights, demonstrating power cancellation in arithmetic progressions and approaching the Ramanujan conjecture bounds.

Contribution

It provides new upper bounds for Kloosterman sum sums over residue classes, advancing understanding of their cancellation properties in number theory.

Findings

Power cancellation in sums over arithmetic progressions

Bounds approaching Ramanujan conjecture

Enhanced understanding of Kloosterman sums in residue classes

Abstract

We prove upper bounds for sums of Kloosterman sums against general arithmetic weight functions. In particular, we obtain power cancellation in sums of Kloosterman sums over arithmetic progressions, which is of square-root strength in any fixed primitive congruence class up to bounds towards the Ramanujan conjecture.