Moments of averages of generalized Ramanujan sums

TL;DR

This paper investigates the distribution of the average values of generalized Ramanujan sums by calculating their moments, providing improved error estimates and extending classical divisor and Ramanujan formulas.

Contribution

It computes the first and second moments of generalized Ramanujan sums with enhanced accuracy and extends related divisor and Ramanujan formulas.

Findings

First and second moments with improved error terms

More accurate main term estimates

Asymptotic results for extended divisor and Ramanujan formulas

Abstract

Let $\beta$ be a positive integer. A generalization of the Ramanujan sum due to Cohen is given by \begin{align} c_{q,\beta }(n) := \sum\limits_{{{(h,{q^\beta })}_\beta } = 1} {{e^{2\pi inh/{q^\beta }}}}, \nonumber \end{align} where $h$ ranges over the 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.