On a criterion for the equality of Dedekind Sums

TL;DR

This paper establishes an equivalence criterion for the equality of Dedekind sums using a modular condition and explores the number of solutions for a given sum when the modulus is square-free.

Contribution

It proves that the equality of Dedekind sums corresponds to a specific integer condition and counts solutions for a fixed sum in the square-free case.

Findings

Dedekind sums are equal iff (m1m2-1)(m1-m2) ≡ 0 mod n

The condition is equivalent to 12s(m1,n)-12s(m2,n) ∈ Z

Number of solutions for fixed m1 when n is square-free is determined

Abstract

In [3] it was shown that the Dedekond sums $s(m_1,n)$ and $s(m_2,n)$ are equal only if $(m_1m_2-1)(m_1-m_2)\equiv 0$ mod $n$. Here we show that the latter condition is equivalent to $12s(m_1,n)-12s(m_2,n)\in \Z$. In addition, we determine, for a given number $m_1$, the number of integers $m_2$ in the range $0\le m_2<n$, $(m_1,m_2)=1$, such that $12s(m_1,n)-12s(m_2,n)\in \Z$, provided that $n$ is square-free.