The number of Dedekind sums with equal fractional parts

TL;DR

This paper investigates the conditions under which Dedekind sums have equal fractional parts and determines the number of solutions for a given parameter, advancing understanding of their modular properties.

Contribution

It provides a formula for counting solutions to a specific congruence related to Dedekind sums, extending previous theoretical results.

Findings

Derived the cardinality of solutions for the congruence $(x-m)(xm-1) mod n$

Established a connection between Dedekind sums and modular arithmetic

Extended previous characterizations of Dedekind sums' fractional parts

Abstract

In a previous it was shown that the Dedkind sums $12s(m,n)$ and $12s(x,n)$, $1\le m,x\le n$, $(m,n)=(x,n)=1$, are equal mod $\Z$ if, and only if, $(x-m)(xm-1)\equiv 0$ mod $n$. Here we determine the cardinality of numbers $x$ in the above range that satisfy this congruence for a given number $m$.