Fix $N$ such that $N\approx 2^{80}$. Let $a,b$ be randomly chosen positive integers such that $a,b<N$ and both $a$ and $b$ are coprime to $N$. I want to show that the odds that $gcd(a,b)=1$ are quite good. I am trying to mimic the argument for random integers, where the odds are $6/\pi^2$. In that argument one uses crucially the fact that, using the natural density measure, the events corresponding to divisibility by various primes are mutually independent. But I am not sure that is even true in this case!
Does anyone know if it is true and if not, is there a workaround?