Moments of averages of generalized Ramanujan sums
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2015
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| _version_ | 1866917563915567104 |
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| author | Robles, Nicolas Roy, Arindam |
| author_facet | Robles, Nicolas Roy, Arindam |
| contents | Let $β$ be a positive integer. A generalization of the Ramanujan sum due to Cohen is given by \begin{align} c_{q,β}(n) := \sum\limits_{{{(h,{q^β})}_β} = 1} {e^{2πinh/{q^β}}}, \nonumber \end{align} where $h$ ranges over the the non-negative integers less than $q^β$ such that $h$ and $q^β$ have no common $β$-th power divisors other than $1$. The distribution of the average value of the Ramanujan sum is a subject of extensive research. In this paper, we study the distribution of the average value of $c_{q,β}(n)$ by computing the $k$-th moments of the average value of $c_{q,β}(n)$. In particular we have provided the first and second moments with improved error terms. We give more accurate results for the main terms than our predecessors. We also provide an asymptotic result for an extension of a divisor problem and for an extension of Ramanujan's formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1508_01760 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Moments of averages of generalized Ramanujan sums Robles, Nicolas Roy, Arindam Number Theory Primary: 11M06, 11N37, Secondary: 11L03, 11N56, 11N64 Let $β$ be a positive integer. A generalization of the Ramanujan sum due to Cohen is given by \begin{align} c_{q,β}(n) := \sum\limits_{{{(h,{q^β})}_β} = 1} {e^{2πinh/{q^β}}}, \nonumber \end{align} where $h$ ranges over the the non-negative integers less than $q^β$ such that $h$ and $q^β$ have no common $β$-th power divisors other than $1$. The distribution of the average value of the Ramanujan sum is a subject of extensive research. In this paper, we study the distribution of the average value of $c_{q,β}(n)$ by computing the $k$-th moments of the average value of $c_{q,β}(n)$. In particular we have provided the first and second moments with improved error terms. We give more accurate results for the main terms than our predecessors. We also provide an asymptotic result for an extension of a divisor problem and for an extension of Ramanujan's formula. |
| title | Moments of averages of generalized Ramanujan sums |
| topic | Number Theory Primary: 11M06, 11N37, Secondary: 11L03, 11N56, 11N64 |
| url | https://arxiv.org/abs/1508.01760 |