Let K be a number field with ring of integers O. After introducing a suitable notion of density for subsets of O, generalising the natural density for subsets of ℤ, we show that the density of the set of coprime m-tuples of algebraic integers is 1/ζK(m), where ζK is the Dedekind zeta function of K. This generalises a result found independently by Mertens ['Ueber einige asymptotische Gesetze der Zahlentheorie', J. reine angew. Math. 77 (1874), 289-338] and Cesàro ['Question 75 (solution)', Mathesis 3 (1883), 224-225] concerning the density of coprime pairs of integers in ℤ.
ON THE MERTENS-CESÀRO THEOREM FOR NUMBER FIELDS
Ferraguti A.
;
2016-01-01
Abstract
Let K be a number field with ring of integers O. After introducing a suitable notion of density for subsets of O, generalising the natural density for subsets of ℤ, we show that the density of the set of coprime m-tuples of algebraic integers is 1/ζK(m), where ζK is the Dedekind zeta function of K. This generalises a result found independently by Mertens ['Ueber einige asymptotische Gesetze der Zahlentheorie', J. reine angew. Math. 77 (1874), 289-338] and Cesàro ['Question 75 (solution)', Mathesis 3 (1883), 224-225] concerning the density of coprime pairs of integers in ℤ.
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