As observed on Mathworld, "Amazingly, the probabilities for random pairs of integers and Gaussian integers being relatively prime are the same as the asymptotic densities of squarefree integers of these types" (respectively, 6/π^{2} and 6/π^{2}K where K is Catalan's constant).

This seems to me unlikely to be a coincidence. Is there a deeper reason - or at least, a proof of this fact which is not equivalent to evaluating both probabilities and observing that they are equal?