On Mertens-Cesàro Theorem for Number Fields

Abstract

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 $1/{\zeta }_K(m)$, where ${\zeta }_K$ is the Dedekind zeta function of $K$.

To appear in the Bulletin of the Australian Mathematical Society

Cited by in corpus (2)

• On the least common multiple of several random integers
• On the logarithmic probability that a random integral ideal is $\mathcal{A}$-free