Kloosterman sums do not correlate with periodic functions

TL;DR

This paper establishes uniform bounds for Kloosterman sums across all arithmetic progressions, demonstrating that these sums do not exhibit correlation with periodic functions, which advances understanding in analytic number theory.

Contribution

The paper provides the first uniform bounds for Kloosterman sums in all arithmetic progressions and proves their lack of correlation with periodic functions.

Findings

Kloosterman sums are uniformly bounded in all arithmetic progressions.

Kloosterman sums do not correlate with periodic functions.

The results have implications for understanding exponential sums in number theory.

Abstract

We provide uniform bounds for sums of Kloosterman sums in all arithmetic progressions. As a consequence, we find that Kloosterman sums do not correlate with periodic functions.