WHEN ARE TWO DEDEKIND SUMS EQUAL?

Abstract

A natural question about Dedekind sums is to find conditions on the integers a1, a2, and b such that s(a1, b) = s(a2, b). We prove that if the former equality holds then b|(a1a2- 1)(a1- a2). Surprisingly, to the best of our knowledge such statements have not appeared in the literature. A similar theorem is proved for the more general Dedekind–Rademacher sums as well, namely that for any fixed non-negative integer n, a positive integer modulus b, and two integers a1and a2that are relatively prime to b, the hypothesis rn(a1, b) = rn(a2, b) implies that b|(6n2+ 1 - a1a2)(a2- a1).