On the probability that two random integers are coprime

TL;DR

The paper explores various probability frameworks under which the probability that two random integers are coprime equals 6/π², including finitely additive, residue class, shift-invariant, and countably additive models.

Contribution

It demonstrates that multiple probability interpretations can support the classical coprimality probability, extending understanding beyond standard models.

Findings

Finitely additive probabilities can assign probability 6/π² to coprimality.

Residue class and shift-invariance models support any probability in [0, 6/π²].

A countably additive probability space supporting 6/π² is constructed.

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 $6/\pi^2$.