On Mertens-Ces\`aro Theorem for Number Fields

TL;DR

This paper generalizes the concept of density for subsets of algebraic integers in a number field and proves that the density of coprime m-tuples equals the reciprocal of the Dedekind zeta function evaluated at m.

Contribution

It introduces a new notion of density for algebraic integers and extends Mertens-Cesàro theorem to arbitrary number fields.

Findings

Density of coprime m-tuples is 1/ζ_K(m)

Generalizes natural density to algebraic integers

Connects algebraic number theory with analytic properties of Dedekind zeta function

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 $K$.