DEDEKIND SUMS WITH SMALL DENOMINATORS

摘要

Let (m,n) = 1 and S(m) = 12s(m), where s(m) is the usual Dedekind sum. Then S(m) 6∈1 Z = {r: r ∈ Z}. Let q > 1 be a divisor of n. We give a necessary and sufficient condition for S(m) G -Z and, thereby, generalize a result of Rademacher that concerns the case q = 1. Further, we study the structure of possible denominators q of S(m). Finally, we show that for certain quadratic irrationals of odd period length l the convergents s_k/t_k, k = l - 1 (mod 2l), yield stationary values of S(s_k/t_k). This means that, for large values of k, the denominators q of these Dedekind sums are very small compared with t_k.