Equality of Dedekind sums mod $\mathbb{Z},2\mathbb{Z}$ and $4\mathbb{Z}$

TL;DR

This paper investigates conditions under which Dedekind sums are equal modulo various integers, providing new proofs and criteria for their equality and exploring their algebraic properties.

Contribution

It offers a new proof of a known criterion for Dedekind sum equality and introduces additional necessary and sufficient conditions for their equality modulo 2 and 4.

Findings

Established that $12s(a_1,b)-12s(a_2,b) otin rac{1}{2}bZ$ and $ otin rac{1}{4}bZ$ unless specific conditions hold.

Provided new necessary conditions for the equality of Dedekind sums modulo 2 and 4.

Enhanced understanding of the algebraic structure of Dedekind sums and their equivalence conditions.

Abstract

In [Girstmair, A criterion for the equality of Dedekind sums mod $\mathbb{Z}$, Internat. J. Number Theory 10: (2014) 565--568], it was shown that the necessary condition $b \mid (a_1 a_2-1)(a_1-a_2)$ for equality of two dedekind sums $s(a_1,b)$ and $s(a_2,b)$ given in [Jabuka, Robins and Wang, When are two Dedekind sums equal? Internat. J. Number Theory 7: (2011) 2197--2202] is equivalent to $12s(a_1,b)-12s(a_2,b) \in \mathbb{Z}$. In this note, we give a new proof of this result and then find two additional necessary and sufficient conditions for $12s(a_1,b)-12s(a_2,b) \in 2\mathbb{Z}, 4\mathbb{Z}$. These give new necessary conditions on equality of Dedekind sums.