Why can we get probability of two integers being coprime while there is no uniform distribution on integers?

Question

I know the coprime probability is about $1/\zeta(2)=6/\pi^2$. However, we can not define uniform distribution on $Z^+\times Z^+$, so how do we define the coprime probability?

Thank you in advance.

Answer 1

The question has already been answered in a comment: $\frac6{\pi^2}$ is the natural density of pairs of coprime integers in $\mathbb N^2$, defined by

$$ \lim_{n\to\infty}\frac{|\{(x,y)\in\mathbb N^2:1\le x,y\le n\land \gcd(x,y)=1\}|}{n^2}\;. $$