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

Using a generalization due to Lerch [M. Lerch, Sur un théorème de Zolotarev. Bull. Intern. de l'Acad. François 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].

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