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Given two randomly chosen positive rational integers, the probability that the two numbers are coprime is $\frac{6}{\pi^2}$. This is also the probability that a positive integer is squarefree. Are there generalizations of these results for Gaussian integers? Or more generally for the ring of integers in an algebraic number field?

edited Jul 3 at 15:58

David Loeffler
8,755●15●40

asked Jul 3 at 14:00

zacarias
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I imagine this has been studied before. The probability ought to be $1/\zeta_K(2)$, where $\zeta_K(s)$ is the Dedekind zeta function of the algebraic number field $K$. – Peter Humphries Jul 3 at 14:16

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One should be careful to specify what model for random integers is used, since there is no canonical probability measure on the integers... – Denis Chaperon de Lauzières Jul 3 at 16:08

Igor Rivin · Accepted Answer · 2012-07-03 14:17:35Z UTC

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There are generalizations, see this mathworld article for some results and references. A detailed exposition for arbitrary number fields is given in this paper by G. Collins and J. Johnson

answered Jul 3 at 14:17

Igor Rivin
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Thank you Igor. – zacarias Jul 3 at 14:32

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This paper deals with the probability that two <i>ideals</i> of norm at most $X$ are coprime, and one needs a small extra argument to deal with principal ideals instead (equidistribution of ideals in ideal classes). – Denis Chaperon de Lauzières Jul 3 at 16:07

I believe the authors comment on the distinction... – Igor Rivin Jul 3 at 18:22

Probability in the Primes

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