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- Title
On Restricted Averages of Dedekind Sums.
- Authors
Minelli, Paolo; Sourmelidis, Athanasios; Technau, Marc
- 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.
- Subjects
DEDEKIND sums; RATIONAL numbers; EUCLIDEAN algorithm
- Publication
IMRN: International Mathematics Research Notices, 2024, Vol 2024, Issue 10, p8485
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnad283