More elementary: the probability that two positive integers have GCD=1 is $6/\pi^2 = 1/\zeta(2)$ because the probability that a prime $p$ divides the GCD is 1/p^2 by considering each p by p block of pairs of positive integers. More generally, the probability that k positive integers have GCD 1 is $1/\zeta(k)$ by a similar argument.