When are two Dedekind sums equal?

Stanislav Jabuka; Sinai Robins; Xinli Wang

paper

When are two Dedekind sums equal?

arXiv:1103.0370

Abstract

A natural question about Dedekind sums is to find conditions on the integers $a_{1}, a_{2}$ , and $b$ such that $s (a_{1}, b) = s (a_{2}, b)$ . We prove that if the former equality holds then $b ∣ (a_{1} a_{2} - 1) (a_{1} - a_{2})$ . 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 $a_{1}$ and $a_{2}$ that are relatively prime to $b$ , the hypothesis $r_{n} (a_{1}, b) = r_{n} (a_{2}, b)$ implies that $b ∣ (6 n^{2} + 1 - a_{1} a_{2}) (a_{2} - a_{1})$ .

6 pages