On the values of Dedekind sums

TL;DR

This paper investigates the possible numerator values of Dedekind sums for a fixed denominator, establishing a residue class structure that simplifies their classification and providing computational results for denominators up to 50.

Contribution

It proves that numerators of Dedekind sums form complete residue classes modulo a specific number, reducing the problem to finitely many residue class representatives and computing these for denominators up to 50.

Findings

Numerators form complete residue classes modulo (q^2-1)q.

All possible numerators for 2 ≤ q ≤ 50 are determined.

A conjecture on all possible Dedekind sum values is proposed.

Abstract

Let $S(a,b)=12s(a,b)$, where $s(a,b)$ denotes the classical Dedekind sum. For a given denominator $q\in \mathbb N$, we study the numerators $k\in\mathbb Z$ of the values $k/q$, $(k,q)=1$, of Dedekind sums $S(a,b)$. Our main result says that if $k$ is such a numerator, then the whole residue class of $k$ modulo $(q^2-1)q$ consists of numerators of this kind. This fact reduces the task of finding all possible numerators $k$ to that of finding representatives for finitely many residue classes modulo $(q^2-1)q$. By means of the proof of this result we have determined all possible numerators $k$ for $2\le q\le 50$, the case $q=1$ being trivial. The result of this search suggests a conjecture about all possible values $k/q$, $(k,q)=1$, of Dedekind sums $S(a,b)$ for an arbitrary $q\in\mathbb N$.