Certain weighted averages of generalized Ramanujan sums

TL;DR

This paper generalizes Ramanujan sums and derives identities involving arithmetical functions, extending Toth's weighted averages by incorporating Cohen's generalized Ramanujan sum.

Contribution

It introduces a new generalization of Ramanujan sums and extends existing weighted average identities to this broader context.

Findings

Derived identities involving arithmetical functions and generalized Ramanujan sums

Extended Toth's weighted averages using Cohen's generalization

Provided new formulas connecting arithmetical functions and generalized sums

Abstract

We derive certain identities involving various known arithmetical functions and a generalized version of Ramanujan sum. L. Toth constructed certain weighted averages of Ramanujan sums with various arithmetic functions as weights. We choose a generalization of Ramanujan sum given by E. Cohen and derive the weighted averages corresponding to the versions of the weighted averages established by Toth.