Kloosterman paths and the shape of exponential sums

TL;DR

This paper studies the distribution of Kloosterman sum paths, proving their convergence to a random Fourier series, and extends results to Birch sums with applications in number theory.

Contribution

It introduces new convergence results for Kloosterman and Birch sum paths using independence of Kloosterman sheaves, advancing understanding of their probabilistic behavior.

Findings

Kloosterman sum paths converge to a random Fourier series

Birch sum paths converge in law in continuous function space

Applications derived from the convergence results

Abstract

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums, as their parameter varies modulo a prime tending to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications.