Averages of Ramanujan sums: Note on two papers by E. Alkan

TL;DR

This paper provides a straightforward proof and a multivariable extension of an identity involving weighted averages of Ramanujan sums, connecting it to various arithmetic functions and special mathematical constants.

Contribution

It introduces a simple proof and a multivariable generalization of Alkan's identity, expanding understanding of Ramanujan sums and their weighted averages.

Findings

Derived identities involving logarithmic weights

Connected Ramanujan sums to Bernoulli polynomials and Gamma function

Extended Alkan's identity to multivariable cases

Abstract

We give a simple proof and a multivariable generalization of an identity due to E. Alkan concerning a weighted average of the Ramanujan sums. We deduce identities for other weighted averages of the Ramanujan sums with weights concerning logarithms, values of arithmetic functions for gcd's, the Gamma function, the Bernoulli polynomials, and binomial coefficients.