DEDEKIND SUMS WITH SMALL DENOMINATORS

作者：Girstmair Kurt^*

来源：International Journal of Number Theory, 2012, 8(8): 1965-1970.

DOI：10.1142/S1793042112501102

• Dedekind sums with small denominators
• stationary value of Dedekind sums

Let (m,n) = 1 and S(m/n) = 12s(m/n), where s(m/n) is the usual Dedekind sum. Then S(m/n) is an element of 1/nZ = {r/n : r is an element of Z}. Let q %26gt;= 1 be a divisor of n. We give a necessary and sufficient condition for S(m/n) is an element of 1/qZ 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 s(k)/t(k), k equivalent to 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).

• 出版日期2012-12