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Feb 22, 2015 at 2:42 comment added Douglas Zare The analogue holds for the probability that $k$ randomly chosen positive integers (up to $n$, then let $n\to \infty$) are relatively prime. J.E Nymann. "On the probability that k positive integers are relatively prime." Journal of Number Theory Volume 4, Issue 5, October 1972, Pages 469–473. A similar result is known for the probability that $k$ positive integers are pairwise coprime.
Feb 22, 2015 at 2:07 vote accept Joseph O'Rourke
Feb 22, 2015 at 2:06 history edited Joseph O'Rourke CC BY-SA 3.0
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Feb 22, 2015 at 2:03 answer added Noam D. Elkies timeline score: 9
Feb 22, 2015 at 1:59 history edited Joseph O'Rourke CC BY-SA 3.0
Answered.
Feb 22, 2015 at 1:02 comment added Gjergji Zaimi Yes, your guess of $1/\zeta(k)$ is correct :) en.wikipedia.org/wiki/…
Feb 22, 2015 at 0:55 history asked Joseph O'Rourke CC BY-SA 3.0