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ON THE MERTENS–CESÀRO THEOREM FOR NUMBER FIELDS

  • ANDREA FERRAGUTI (a1) and GIACOMO MICHELI (a2)
Abstract

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 $\mathbb{Z}$ .

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Corresponding author
giacomo.micheli@math.uzh.ch
References
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[1]Barroero, F., Frei, C. and Tichy, R., ‘Additive unit representations in rings over global fields – a survey’, Publ. Math. Debrecen 79(3) (2011), 291307.
[2]Cellarosi, F. and Vinogradov, I., ‘Ergodic properties of k-free integers in number fields’, J. Mod. Dyn. 7(3) (2013), 461488.
[3]Cesàro, E., ‘Question proposée 75’, Mathesis 1 (1881), 184.
[4]Cesàro, E., ‘Question 75 (solution)’, Mathesis 3 (1883), 224225.
[5]Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers (Clarendon Press, Oxford, 1960).
[6]Masser, D. and Vaaler, J. D., ‘Counting algebraic numbers with large height II’, Trans. Amer. Math. Soc. 359(1) (2007), 427445.
[7]Maze, G., Rosenthal, J. and Wagner, U., ‘Natural density of rectangular unimodular integer matrices’, Linear Algebra Appl. 434(5) (2011), 13191324.
[8]Mertens, F., ‘Ueber einige asymptotische Gesetze der Zahlentheorie’, J. reine angew. Math. 77 (1874), 289338.
[9]Nymann, J. E., ‘On the probability that k positive integers are relatively prime’, J. Number Theory 4(5) (1972), 469473.
[10]Schanuel, S., ‘Heights in number fields’, Bull. Soc. Math. France 107(4) (1979), 433449.
[11]Sittinger, B. D., ‘The probability that random algebraic integers are relatively r-prime’, J. Number Theory 130(1) (2010), 164171.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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