The largest values of Dedekind sums

TL;DR

This paper characterizes the largest values of Dedekind sums for large moduli, identifying a specific set of integers where the sums reach or exceed certain bounds, thus advancing understanding of their extremal behavior.

Contribution

It provides a precise description of the integers that maximize Dedekind sums relative to a fixed parameter, revealing the structure of their largest values for large moduli.

Findings

Identifies at most k^2 integers where Dedekind sums are maximal

Shows for most integers, Dedekind sums are less than a specific bound

Provides bounds for Dedekind sums for large n

Abstract

Let $s(m,n)$ denote the classical \DED sum, where $n$ is a positive integer and $m\in\{0,1,\ldots, n-1\}$, $(m,n)=1$. For a given positive integer $k$, we describe a set of at most $k^2$ numbers $m$ for which $s(m,n)$ may be $\ge s(k,n)$, provided that $n$ is sufficiently large. For the numbers $m$ not in this set, $s(m,n)<s(k,n)$.