  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.
