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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.

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    $\begingroup$ I saw a proof of this somewhere using the zeta function. I'll try and dig it up. EDIT: Here it is: mathreference.com/lc-z,cop.html. $\endgroup$ Jul 27, 2015 at 12:52
  • $\begingroup$ OP: Can you prove the main ingredient the proof in the accepted answer relies on (that divisibilities by different primes are independent events)? $\endgroup$
    – Did
    Aug 30, 2015 at 20:31
  • $\begingroup$ Yes I can prove this. $\endgroup$
    – apkg
    Sep 1, 2015 at 16:20
  • $\begingroup$ Then adding this proof would make the page much more useful to its potential readers. (Unrelated: Please use @user to signal a comment to user.) $\endgroup$
    – Did
    Sep 2, 2015 at 21:14

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Two integers are coprime iff there is no prime $p$ dividing both of them, so the asymptotic probability is: $$ \frac{1}{2}\prod_{p>2}\left(1-\frac{1}{p^2}\right)=\frac{2}{3}\prod_{p}\left(1-\frac{1}{p^2}\right)=\frac{2}{3\zeta(2)}=\color{red}{\frac{4}{\pi^2}}.$$

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  • $\begingroup$ ...Assuming that divisibilities by different primes are independent events--which is kind of the main part of the proof. $\endgroup$
    – Did
    Aug 28, 2015 at 13:58

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