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ON THE MERTENS–CESÀRO THEOREM FOR NUMBER FIELDS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 93 / Issue 2 / April 2016
- Published online by Cambridge University Press:
- 11 November 2015, pp. 199-210
- Print publication:
- April 2016
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- Article
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- You have access
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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 
$\mathbb{Z}$.
