On Restricted Averages of Dedekind Sums

Abstract

Abstract We investigate the averages of Dedekind sums over rational numbers in the set $\mathscr {F}_{\alpha }(Q) = \{\,{v}/{w}\in \mathbb {Q}: 0<w\leq Q\,\}\cap \lbrack 0, \alpha )$ for fixed $\alpha \leq 1/2$. In previous work, we obtained asymptotics for $\alpha =1/2$, confirming a conjecture of Ito in a quantitative form. In the present article we extend our former results, first to all fixed rational $\alpha $ and then to almost all irrational $\alpha $. As an intermediate step we obtain a result quantifying the bias occurring in the second term of the asymptotic for the average running time of the by-excess Euclidean algorithm, which is of independent interest.