I found the answer to a very similar question already asked here on mathoverflow: what is the probability that two natural numbers are relatively prime? The answer given in the link below was $\frac{6}{\pi^2}$.
What is the probability that two numbers are relatively prime?
My question is a little more specific: if two integers are randomly selected from the interval $[a,b]$, what is the probability that they are relatively prime?
I'd guess that as $a$ and $b$ get farther apart, the probability would approach the same $\frac{6}{\pi^2}$. I'm trying to find an answer for when $a$ and $b$ are quite large, say around $2^{128}$, but I'd love to see an analysis for general integers. Since the similar question was already answered above, I wouldn't think it would be much work to solve the problem on a smaller interval.
