The denominator is simple, just n^2.

The numerator is: \sum_{k=1}^n \phi(k)\cdot \left[ \frac{n}{k} \right] + a(n,k).

Where [ ] is the greatest integer function.

And a(n,k) is the number of integers in the interval [1,n*mod(k)] relatively prime to k.

The problem with this formula is that it is not pleasant to use in an limit. Another problem is a(n,k). Thus, my idea is going to produce is good lower bound and upper bound for this sum. The first bound to try is to use the fact that 1<=a(n,k)<=(k-1). But I am not sure if this is a good approximation. Another approximation is on phi(k) which can be found here.