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Moments of averages of generalized Ramanujan sums

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Abstract

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.

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Acknowledgments

The first author wishes to acknowledge partial support of SNF Grant 200020-\(149150 \backslash 1\). The authors are grateful to the referee for corrections and helpful remarks.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL, 61801, USA

    Nicolas Robles

  2. Mathematics Department, Rice University, MS 136, 6100 Main Street, Houston, TX, 77005, USA

    Arindam Roy

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  1. Nicolas Robles
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  2. Arindam Roy
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Correspondence to Arindam Roy.

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Communicated by A. Constantin.

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Robles, N., Roy, A. Moments of averages of generalized Ramanujan sums. Monatsh Math 182, 433–461 (2017). https://doi.org/10.1007/s00605-016-0907-z

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