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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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 

