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 $n\rightarrow\infty$.



What is the probability that two numbers are relatively prime?
The probability tends to $\frac{1}{\zeta(2)}=\frac{6}{\pi^2}$ as was mentioned by Qiaochu. This actually generalizes to arbitrary number fields, and is a less commonly known fact.


In fact in any number field, the probability that two ideals are relatively prime is given by $1/\zeta\_K(2)$, where $\zeta\_K$ is the [Dedekind zeta function](https://en.wikipedia.org/wiki/Dedekind_zeta_function) of the number field $K$. And is proven in a similar way to the classical result. Here is a reference: ["The probability of relative primality of Gaussian integers"](https://doi.org/10.1007/3-540-51084-2_23). For example the analogous probability for Gaussian integers is $6/(\pi^2G)$ where $G=1-\frac{1}{3^2}+\frac{1}{5^2}+\cdots$ is the [Catalan constant](https://en.wikipedia.org/wiki/Catalan_constant).


