Mathworld article - Gaussian Integers - Catalan's constant - ??

Questions
1

The statement says “The probability that two Gaussian integers (a) and (b) are relatively prime is … (math)”

It seems bogus since, as on the integers, there’s not a uniform distribution on the Gaussian Integers. What probability distribution is it talking about?

Seems bogus. Can someone with access to (Pegg; Collins and Johnson 1989; Finch 2003, p. 601) make some sense of that for me? Maybe it could be written better: I did not see a feedback link on the page.

It’s not for research. I am not likely to be very interested in the technical answer, but I seem to be interested in the subject of Gaussian integers.

http://mathworld.wolfram.com/GaussianInteger.html

2

The statement should be understood in an asymptotic sense, just as its analog for integers (the probability that any two randomly chosen integers are relatively prime is 6/Pi^2).

For an illustration, consider a simpler question, what is the probability that a randomly chosen integer is divisible by 5? Among the numbers {1, 2, 3, …, n}, there are Floor[n/5] multiples of 5, so the probability to choose one at random is Floor[n/5]/n. The limit of that probability as n goes to Infinity is 1/5.

I don’t have a link to the full Collins paper, but here is a preview. Also, see this paper for another proof.

The following experiment generates 10^5 pairs of Gaussian integers and compares the empirical probability to the theoretical value:

In[6]:= r:=RandomInteger[{-2^128, 2^128}, {10^5, 2}]; N[Total[Boole[CoprimeQ @@@ (r + I r)]/10^5]]

Out[6]= 0.66395

In[7]:= 6./(Pi^2 Catalan)

Out[7]= 0.663701