Another generalization of Euler's arithmetic function and Menon's identity

TL;DR

This paper introduces a new generalized Euler function for k-tuples, explores its properties, and derives a Menon-type identity, extending classical number theory concepts to higher dimensions.

Contribution

It defines the k-dimensional generalized Euler function and establishes a new Menon-type identity related to it, expanding the scope of classical arithmetic functions.

Findings

Properties of the generalized Euler function $\

$ ext{ and its behavior analyzed.

A new Menon-type identity involving $\

Abstract

We define the $k$-dimensional generalized Euler function $\varphi_k(n)$ as the number of ordered $k$-tuples $(a_1,\ldots,a_k)\in {\Bbb N}^k$ such that $1\le a_1,\ldots,a_k\le n$ and both the product $a_1\cdots a_k$ and the sum $a_1+\cdots +a_k$ are prime to $n$. We investigate some of properties of the function $\varphi_k(n)$, and obtain a corresponding Menon-type identity.