On the equality of Dedekind sums

TL;DR

This paper investigates the conditions under which Dedekind sums are equal, providing a method to determine equality, constructing infinite equal sum sequences, and analyzing the number of equal sums for square-free moduli.

Contribution

It establishes an equivalence between Dedekind sum equality and a value decision problem, introduces infinite sequences of equal sums, and bounds the number of equal sums for square-free moduli.

Findings

Deciding Dedekind sum equality is equivalent to a specific value decision.

Constructed infinite sequences of pairwise equal Dedekind sums.

Bounded the number of equal Dedekind sums for square-free moduli.

Abstract

We show that deciding the equality of two Dedekind sums $S(c,b)$, $S(d,b)$ is equivalent to deciding whether a Dedekind sum defined by $b, c, d$ takes a certain value. By means of this result we construct infinite sequences of pairwise equal Dedekind sums. Moreover, we prove a result that says how many Dedekind sums $S(d,b)$, $1\le d\le b-1$, may be equal to a given $S(c,b)$ if $b$ is a square-free number.