Moments of averages of generalized Ramanujan sums
详细信息 查看全文
- 作者：Nicolas Robles ; Arindam Roy
- 关键词：Generalized Ramanujan sum ; Ramanujan’s formula ; Divisor problem ; Perron’s formula ; Moments estimates ; Asymptotic formulas
- 刊名：Monatshefte für Mathematik
- 出版年：2017
- 出版时间：February 2017
- 年：2017
- 卷：182
- 期：2
- 页码：433-461
- 全文大小：
- 刊物主题：Mathematics, general;
- 出版者：Springer Vienna
- ISSN：1436-5081
- 卷排序：182
文摘
Let \(\beta \) be a positive integer. A generalization of the Ramanujan sum due to Cohen is given by $$\begin{aligned} c_{q,\beta }(n) := \sum \limits _{{{(h,{q^\beta })}_\beta } = 1} {{e^{2\pi inh/{q^\beta }}}}, \end{aligned}$$where h ranges over the non-negative integers less than \(q^{\beta }\) such that h and \(q^{\beta }\) have no common \(\beta \)-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 \(c_{q,\beta }(n)\) by computing the k-th moments of the average value of \(c_{q,\beta }(n)\). 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.KeywordsGeneralized Ramanujan sumRamanujan’s formulaDivisor problemPerron’s formulaMoments estimatesAsymptotic formulasCommunicated by A. Constantin.