Abstract
Let (m, n) = 1 and S(m/n) = 12s(m/n), where s(m/n) is the usual Dedekind sum. Then [Formula: see text]. Let q ≥ 1 be a divisor of n. We give a necessary and sufficient condition for [Formula: see text] 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/n). Finally, we show that for certain quadratic irrationals of odd period length l the convergents sk/tk, k ≡ l - 1 ( mod 2l), yield stationary values of S(sk/tk). This means that, for large values of k, the denominators q of these Dedekind sums are very small compared with tk.
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