On restricted averages of Dedekind sums
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915129937887232 |
|---|---|
| author | Minelli, Paolo Sourmelidis, Athanasios Technau, Marc |
| author_facet | Minelli, Paolo Sourmelidis, Athanasios Technau, Marc |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_00441 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On restricted averages of Dedekind sums Minelli, Paolo Sourmelidis, Athanasios Technau, Marc Number Theory 11A55, 11F20, 11K50, 11J25 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. |
| title | On restricted averages of Dedekind sums |
| topic | Number Theory 11A55, 11F20, 11K50, 11J25 |
| url | https://arxiv.org/abs/2301.00441 |