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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?

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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 '12 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 '12 at 16:08

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up vote 9 down vote accepted

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

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Thank you Igor. –  zacarias Jul 3 '12 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 '12 at 16:07
    
I believe the authors comment on the distinction... –  Igor Rivin Jul 3 '12 at 18:22

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