On the Probability That Two Random Integers Are Coprime

Abstract

We show that there is a nonempty 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 $6/\pi ^{2}$.