Equality of Dedekind sums modulo 24\mathbb Z

2016/09/27 by Kurt Girstmair, Girstmair, Kurt
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1609.08282

arxiv created 2016/09/27 · arxiv updated 2016/09/28

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(a₁,b)-S(a₂,b)∈ 8\mathbb Z. In the present paper we show that this condition is equivalent to S(a₁,b)-S(a₂,b)∈ 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| b.

Related

• Congruences mod 4 for the alternating sum of the partial quotients
• On the moduli of a Dedekind sum
• On a criterion for the equality of Dedekind Sums
• On the values of Dedekind sums
• Equality of Dedekind sums mod 8 ℤ