On restricted averages of Dedekind sums

TL;DR

The paper advances the understanding of restricted averages of Dedekind sums by analyzing sums over Farey fractions up to $Q$ with a cut at $\alpha$. For rational $\alpha=v/w$ in $(0,1/2]$, the authors prove the leading term $\frac{1}{12}\bigl(1-\frac{1}{w^2}\bigr)\log Q$ and establish bounded errors $O_\alpha(1)$, extending Ito’s conjecture beyond $\alpha=1/2$. For almost all irrational $\alpha$, they show a full-measure set $\mathscr{S}$ in $[0,1/2]$ where the average has leading term $\frac{1}{12}\log Q$ with a subleading, $o_\alpha(\log Q)$ error, reflecting a bias driven by near-zero Farey fractions. A key technical component is a main proposition counting solutions to Diophantine systems tied to modular inverses and minus continued fractions, combined with Möbius inversion and auxiliary asymptotic tools, which also yields insights into the by-excess Euclidean algorithm’s running time bias. The results link Dedekind sums, continued fractions, and Euclidean algorithms, delivering precise asymptotics and highlighting the role of rational versus irrational cuts in the distribution of $s(x)$.

Abstract

We investigate the averages of Dedekind sums over rational numbers in the set $$\mathscr{F}_α(Q):=\{\, {v}/{w}\in \mathbb{Q}: 0<w\leq Q\,\}\cap [0, α)$$ for fixed $α\leq 1/2$. In previous work, we obtained asymptotics for $α=1/2$, confirming a conjecture of Ito in a quantitative form. In the present article we extend our former results, first to all fixed rational $α$ and then to almost all irrational $α$. 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.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4