Moments of averages of generalized Ramanujan sums
Moments of averages of generalized Ramanujan sums
复制标题
广义拉马努金和的平均矩
DOI：
10.1007/s00605-016-0907-z
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Arindam Roy
中科院分区：
文献类型：
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作者：
Nicolas Robles;Arindam Roy
设β是一个正整数。由Cohen引起的Ramanujan和的一个推广是c_ q， β (n):=∑\极限_ (h, q^ β) _ β= 1e ^ 2 π inh/q^ β, cq, β (n):=∑(h, q β) β= 1e ^ 2 π inh/q β，其中h的取值范围小于q^ β q β的非负整数，使得h和q^ β q β除了1之外没有公共的β β-幂因子。拉马努金和平均值的分布是一个广泛研究的课题。本文通过计算c_ q， β (n) cq, β (n) cq, β (n)的平均值的k阶矩，研究了c_ q， β (n) cq, β (n)的平均值的分布。特别地，我们为第一阶矩和第二阶矩提供了改进的误差项。与前人相比，我们对主要项给出了更准确的结果。我们也给出了一个除数问题的推广和拉马努金公式的推广的渐近结果。
Let β β be a positive integer. A generalization of the Ramanujan sum due to Cohen is given by c_ q, β (n):= ∑\limits _ (h, q^ β) _ β= 1 e^ 2 π inh/q^ β, cq, β (n):=∑(h, q β) β= 1 e 2 π inh/q β, where h ranges over the non-negative integers less than q^ β q β such that h and q^ β q β have no common β 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) cq, β (n) by computing the k-th moments of the average value of c_ q, β (n) cq, 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.