The Limiting Distribution of Character Sums

TL;DR

This paper investigates the distribution of Dirichlet character sums modulo prime q, establishing a limiting distribution as q approaches infinity through the use of Steinhaus random multiplicative functions, inspired by prior work on Kloosterman paths.

Contribution

It introduces a new limiting distribution for Dirichlet character sums using Steinhaus functions, extending understanding of their complex path behavior.

Findings

Identifies the limiting distribution of character sum paths as q approaches infinity.

Characterizes properties of the limiting random process.

Connects the distribution to Steinhaus random multiplicative functions.

Abstract

In this paper, we consider the distribution of the continuous paths of Dirichlet character sums modulo prime $q$ on the complex plane. We also find a limiting distribution as $q \rightarrow \infty$ using Steinhaus random multiplicative functions, stating properties of this random process. This is motivated by Kowalski and Sawin's work on Kloosterman paths.