On restricted averages of Dedekind sums

TL;DR

This paper extends previous results on the averages of Dedekind sums over rational numbers, now including all fixed rational and almost all irrational bounds, and also investigates bias in the Euclidean algorithm's running time.

Contribution

It generalizes earlier asymptotic results for Dedekind sums to all fixed rational and almost all irrational bounds, and quantifies bias in the Euclidean algorithm's second-term asymptotics.

Findings

Asymptotics confirmed for all fixed rational lpha.

Extension of results to almost all irrational lpha.

Quantification of bias in Euclidean algorithm's second term.

Abstract

We investigate the averages of Dedekind sums over rational numbers in the set $$\mathscr{F}_\alpha(Q):=\{\, {v}/{w}\in \mathbb{Q}: 0<w\leq Q\,\}\cap [0, \alpha)$$ 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 \textit{by-excess} Euclidean algorithm, which is of independent interest.