On Dedekind sums with equal values

Abstract

Dedekind sums $s(m,n)$ occur in many fields of mathematics. Since $s(m_1,n)=s(m_2,n)$ if $m_1\equiv m_2$ mod $n$, it is natural to ask which of the Dedekind sums $s(m,n)$, $0\le m<n$, take equal values. So far no simple criterion is known by which the equality of $s(m_1,n)$ and $s(m_2,n)$ could be decided. In this note we show how to obtain non-obvious examples of equal Dedekind sums. We consider two cases which mark the extreme possibilities for the argument $n$, namely, $n$ a prime power and $n$ square-free. Whereas we can give a partial overview of equal Dedekind sums in the prime power case, such an overview seems to be much more difficult to obtain in the square-free case.