The basic question that I have is in the title, but let us make it more rigorous below.

Let $N=\{1, 2, ..., n\}$, and put the (normalized) counting measure, $\mu_n$, on $N\times N$.

Let $\mathcal{S}_n= \{ (a, b)\in N\times N: gcd(a, b)=1\}$

and $x_n=\mu_n(\mathcal{S}_n).$

Then what is the assymptotic behavior of $x_n$ as $n\rightarrow\infty$.