Equality of Dedekind sums mod $8 \mathbb{Z}$

TL;DR

This paper establishes precise conditions under which the difference of two Dedekind sums is divisible by 8, using a generalized lemma related to quadratic reciprocity, and confirms a conjecture about partial quotient sums.

Contribution

It provides new necessary and sufficient conditions for Dedekind sums to differ by multiples of 8 and resolves a conjecture on partial quotient sums.

Findings

Necessary and sufficient conditions for Dedekind sums difference in 8ℤ.

New necessary conditions for equality of Dedekind sums.

Resolution of Girstmair's conjecture on partial quotient sums.

Abstract

Using a generalization due to Lerch [M. Lerch, Sur un th\'{e}or\`{e}me de Zolotarev. Bull. Intern. de l'Acad. Fran\c{c}ois Joseph 3 (1896), 34-37] of a classical lemma of Zolotarev, employed in Zolotarev's proof of the law of quadratic reciprocity, we determine necessary and sufficient conditions for the difference of two Dedekind sums to be in $8\mathbb{Z}$. These yield new necessary conditions for equality of two Dedekind sums. In addition, we resolve a conjecture of Girstmair [Girstmair, Congruences mod 4 for the alternating sum of the partial quotients, arXiv: 1501.00655].