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THE DENSITY OF $j$ -WISE RELATIVELY $r$ -PRIME ALGEBRAIC INTEGERS

  • BRIAN D. SITTINGER (a1)

Abstract

Let $K$ be a number field with a ring of integers ${\mathcal{O}}$ . We follow Ferraguti and Micheli [‘On the Mertens–Cèsaro theorem for number fields’, Bull. Aust. Math. Soc.93(2) (2016), 199–210] to define a density for subsets of ${\mathcal{O}}$ and use it to find the density of the set of $j$ -wise relatively $r$ -prime $m$ -tuples of algebraic integers. This provides a generalisation and analogue for several results on natural densities of integers and ideals of algebraic integers.

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[1] Benkoski, S. J., ‘The probability that k positive integers are relatively r-prime’, J. Number Theory 8 (1976), 218223.
[2] DeMoss, R. D. and Sittinger, B. D., ‘The probability that $k$ ideals in a ring of algebraic integers are $m$ -wise relatively prime’, Preprint, 2018, arXiv:1803.09187.
[3] Ferraguti, A. and Micheli, G., ‘On the Mertens–Cèsaro theorem for number fields’, Bull. Aust. Math. Soc. 93(2) (2016), 199210.
[4] Gegenbauer, L., ‘Asymptotische Gesetze der Zahlentheorie’, Denkshcriften Akad. Wien 9 (1885), 3780.
[5] Hu, J., ‘The probability that random positive integers are k-wise relatively prime’, Int. J. Number Theory 9 (2013), 12631271.
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[7] Mertens, F., ‘Über einige asymptotische Gesetze der Zahlentheorie’, J. reine angew. Math. 77 (1874), 289338.
[8] Sittinger, B. D., ‘The probability that random algebraic integers are relatively r-prime’, J. Number Theory 130 (2010), 164171.
[9] Tóth, L., ‘The probability that k positive integers are pairwise relatively prime’, Fibonacci Quart. 40 (2002), 1318.
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THE DENSITY OF $j$ -WISE RELATIVELY $r$ -PRIME ALGEBRAIC INTEGERS

  • BRIAN D. SITTINGER (a1)

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