On the probability that two random integers are coprime

Abstract

We show that there is a non-empty class of finitely additive probabilities on $\mathbb N^2$ such that for each member of the class, each set with limiting relative frequency $p$ has probability $p$. Hence, in that context the probability that two random integers are coprime is $6/\pi^2$. We also show that two other interpretations of "random integer," namely residue classes and shift invariance, support any number in $[0, 6/\pi^2]$ for that probability. Finally, we specify a countably additive probability space that also supports $6/\pi^2$.