ON THE MERTENS–CESÀRO THEOREM FOR NUMBER FIELDS
2015 ◽
Vol 93
(2)
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pp. 199-210
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Keyword(s):
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 $\mathbb{Z}$.
1995 ◽
Vol 138
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pp. 199-208
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2013 ◽
Vol 12
(1)
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pp. 137-165
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2018 ◽
Vol 98
(2)
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pp. 221-229
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2019 ◽
Vol 19
(04)
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pp. 2050080
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2012 ◽
Vol 08
(01)
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pp. 125-147
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2017 ◽
Vol 147
(2)
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pp. 245-262
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