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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$\begingroup$ 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 $K$. $\endgroup$– Peter HumphriesCommented Jul 3, 2012 at 14:16
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3$\begingroup$ One should be careful to specify what model for random integers is used, since there is no canonical probability measure on the integers... $\endgroup$– Denis Chaperon de LauzièresCommented Jul 3, 2012 at 16:08
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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, 2012 at 14:17
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2$\begingroup$ 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). $\endgroup$ Commented Jul 3, 2012 at 16:07
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$\begingroup$ I believe the authors comment on the distinction... $\endgroup$ Commented Jul 3, 2012 at 18:22