On restricted averages of Dedekind sums

Abstract

We investigate the averages of Dedekind sums over rational numbers in the set $${\mathcal{F}}_{\hat{I}}\pm (Q):=\{v/w\in \mathbb{Q}:0<w\le Q\}\cap [0,\hat{I}\pm )$$ for fixed $\hat{I}\pm \le 1/2$. In previous work, we obtained asymptotics for $\hat{I}\pm =1/2$, confirming a conjecture of Ito in a quantitative form. In the present article we extend our former results, first to all fixed rational $\hat{I}\pm$ and then to almost all irrational $\hat{I}\pm$. 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 \textit{by-excess} Euclidean algorithm, which is of independent interest.

21 pages. arXiv admin note: text overlap with arXiv:2206.03214