I'm having difficulty with a number-theory-type exercise. Could you provide assistance with computing the asymptotic probabilities that two integers are coprime (both integers tending to $\infty$), given that their maximum is even?
I have essentially no experience in number theory and have been asked this by a colleague, so I thought I'd pass it over here.
Two integers are coprime iff there is no prime $p$ dividing both of them, so the asymptotic probability is: $$ \frac{1}{2}\prod_{p>2}\left(1-\frac{1}{p^2}\right)=\frac{2}{3}\prod_{p}\left(1-\frac{1}{p^2}\right)=\frac{2}{3\zeta(2)}=\color{red}{\frac{4}{\pi^2}}.$$