Another generalization of Euler\u2019s arithmetic function and Menon\u2019s identity

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 {mathbb {N}}^k such that 1le a_1,ldots ,a_kle n and both the product a_1cdots a_k and the sum a_1+cdots +a_k are prime to n. We investigate some of the properties of the function varphi _k(n), and obtain a corresponding Menon-type identity.