On Mertens-Ces\`aro Theorem for Number Fields

Let $K$ be a number field with ring of integers $\mathcal{O}$. After introducing a suitable notion of density for subsets of $\mathcal{O}$, generalizing that of natural density for subsets of $\mathbb{Z}$, we show that the density of the set of coprime $m$-tuples of algebraic integers is $1\mathrm{/}{\zeta }_K(m)$, where ${\zeta }_K$ is the Dedekind zeta function of $K$.