EQUALITY OF DEDEKIND SUMS MODULO 24Z

摘要

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 sucient condition for S(a1, b) S(a2, b) 2 8Z. In the present paper we show that this condition is equivalent to S(a1, b) S(a2, b) 2 24Z, provided that 9 - 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 - b and mod 72 if 3 | b.