# Probability in the Primes

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
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

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