On the distribution of Dedekind sums

TL;DR

This paper surveys various aspects of Dedekind sums, including their distribution, density, mean values, and behavior near quadratic irrationals, highlighting recent results and open questions in the field.

Contribution

It provides a comprehensive overview of the distribution properties of Dedekind sums, supplementing previous surveys with recent findings and detailed analysis.

Findings

Dedekind sums are densely distributed in certain intervals.

They exhibit uniform distribution under specific conditions.

Behavior near quadratic irrationals shows distinctive patterns.

Abstract

Dedekind sums have applications in quite a number of fields of mathematics. Therefore, their distribution has found considerable interest. This article gives a survey of several aspects of the distribution of these sums. In particular, it highlights results about the values of Dedekind sums, their density and uniform distribution. Further topics include mean values, large and small (absolute) values, and the behaviour of Dedekind sums near quadratic irrationals. The present paper can be considered as a supplement to the survey article [R. W. Bruggeman, On the distribution of Dedekind sums, Contemp. Math. 166 (1994), 197--210].