# Expansions of arithmetic functions of several variables with respect to certain modified unitary Ramanujan sums

## TL;DR

This paper introduces new Ramanujan sum analogues based on unitary divisors and explores their use in expanding multivariable arithmetic functions, with applications to unitary sigma and phi functions.

## Contribution

It presents novel modified Ramanujan sums related to unitary divisors and develops expansion formulas for multivariable arithmetic functions using these sums.

## Key findings

- Derived new expansion formulas for arithmetic functions of several variables.
- Applied results to functions involving unitary sigma and phi functions.
- Established properties of the new Ramanujan sum analogues.

## Abstract

We introduce new analogues of the Ramanujan sums, denoted by $\widetilde{c}_q(n)$, associated with unitary divisors, and obtain results concerning the expansions of arithmetic functions of several variables with respect to the sums $\widetilde{c}_q(n)$. We apply these results to certain functions associated with $\sigma^*(n)$ and $\phi^*(n)$, representing the unitary sigma function and unitary phi function, respectively.