Equality of Dedekind sums modulo 24$\mathbb Z$

TL;DR

This paper investigates the divisibility properties of Dedekind sums, establishing new conditions for their equality modulo 24, and extends known congruences related to continued fraction partial quotients.

Contribution

It proves that the condition for Dedekind sums difference to be divisible by 8 is equivalent to divisibility by 24 when 9 does not divide the denominator, and refines congruences for partial quotient sums.

Findings

Dedekind sum differences are divisible by 24 under certain conditions.

Extended congruences for partial quotients modulo 24 and 72.

Established equivalence of divisibility conditions for Dedekind sums.

Abstract

Let $S(a,b)=12s(a,b)$, where $s(a,b)$ denotes the classical Dedekind sum. In a recent note E. Tsukerman gave a necessary and sufficient condition for $S(a_1,b)-S(a_2,b)\in 8\mathbb Z$. In the present paper we show that this condition is equivalent to $S(a_1,b)-S(a_2,b)\in 24\mathbb Z$, provided that $9\nmid b$. Tsukerman also obtained a congruence mod 8 for $bT(a,b)$, where $T(a,b)$ is the alternating sum of the partial quotients of the continued fraction expansion of $a/b$. We show that the respective congruence holds mod $24$ if $3\nmid b$ and mod $72$ if $3\mid b$.