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we all know that if we pick two random integer numbers, then the probability of them being coprimes is $6/\pi^2$.

But as I have understood that probability is true if we pick a number that goes from $0$ to $\infty$. What is that probability if we pick a number that goes from $0$ to $N$ where $N$ is an integer positive number ?

Davinator
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    The standard proofs of the claim you start with finite bounds and pass to the limit. See, e.g., this. – lulu Sep 14 '20 at 12:54
  • Thanks Lulu, and sorry for my lack of "pro" knowledge. How can I derive a formula that does not pass to the limit ? A formula that involves the term N , for example ? – Davinator Sep 14 '20 at 13:10
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    The link i provided has formulas for a finite bound (they use $x$ instead of $N$ to denote the bound). the exact formulas are, of course, a bit hard to work with but the asymptotic formulas are good. – lulu Sep 14 '20 at 13:18

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