Probability for two randomly chosen positive integers to have no common factors

We have previously looked at the evaluation of the Riemann zeta function at 2:

This result can be used to calculate the probability for two randomly chosen positive integers to have no common factors.  It is easy to see that the probability for two randomly chosen positive integers to be both multiples of a prime p is 1/p².  So the probability that they do not share a common factor of p is 1 − 1/p².  Therefore the probability that they do not have ANY common factors is:

Very neat result!