TITLE: On the probability that two positive integers are relatively prime
QUESTION [12 upvotes]: In many of the sources I have consulted about this, the "probability" that two positive integers chosen at random are relatively prime is calculated as the limit as $n \to \infty$ of the probability that two randomly chosen integers in the set {1,2, ..., $n$} are relatively prime (the limit being $1/\zeta(2)$). My first question is: Is this limit really a probability?
Also, the nonrigorous/heuristic proofs that I have seen of this start by mentioning that "the probability that a prime $p$ divides a positive integer is $1/p$". This makes intuitive sense. I was wondering though: Is there a way of defining a probability measure on the positive integers in such a way that the set {$n \in \mathbf Z_+$ | $p$ divides $n$} has measure $1/p$ (that we can use for a rigorous proof)?

REPLY [3 votes]: One may use the honest probability $P(X=n)=n^{-s}/\zeta(s)$ for $s>1$.
From this honest probability one may prove that:
1 Events $E_{p}=\{X$ is divisible by $p\}$ are independent.
2 $P(gcd(X,Y)=n)=n^{-2s}/\zeta(2s)$ for independent $X$ and $Y$. If $s\to 1$ for $n=1$ we obtain the claim.
Some other results may be obtained. See ex. 4.2. in Probability with martingales by Williams.