On Mertens-Ces\`aro Theorem for Number Fields
Andrea Ferraguti, Giacomo Micheli

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 be a number field with ring of integers . After introducing a suitable notion of density for subsets of , generalizing that of natural density for subsets of , we show that the density of the set of coprime -tuples of algebraic integers is , where is the Dedekind zeta function of .
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