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ON THE MERTENS–CESÀRO THEOREM FOR NUMBER FIELDS

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  11 November 2015

ANDREA FERRAGUTI  and
GIACOMO MICHELI
ANDREA FERRAGUTI
Affiliation:
Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland email andrea.ferraguti@math.uzh.ch
GIACOMO MICHELI*
Affiliation:
Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland email giacomo.micheli@math.uzh.ch
*
✉giacomo.micheli@math.uzh.ch
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Abstract

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

Keywords

number fieldsalgebraic integersnatural densityMertens–Cesàro theoremzeta function

MSC classification

Primary: 11R45: Density theorems
Secondary: 11R04: Algebraic numbers; rings of algebraic integers
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 2 , April 2016 , pp. 199 - 210
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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