Prime-Residue-Class of Uniform Charges on the Integers

Abstract

There is a probability charge on the power set of the integers that gives probability $1/p$ to every residue class modulo a prime $p$. There exists such a charge that gives probability $w$ to the set of prime numbers iff $w \in [0,1/2]$. Similarly, there is such a charge that gives probability $x$ to a residue class modulo $c$, where $c$ is composite, iff $x \in [0,1/y]$, where $y$ is the largest prime factor of $c$.