Prime-Residue-Class of Uniform Charges on the Integers

TL;DR

This paper explores probability charges on integers assigning uniform probabilities to residue classes, revealing conditions under which specific probabilities are achievable for primes and composite moduli.

Contribution

It introduces a novel class of probability charges on integers that assign uniform probabilities to residue classes and characterizes their possible values for primes and composites.

Findings

Probability charge assigns 1/p to each residue class mod p.

Charge assigns probability w to primes iff w ∈ [0, 1/2].

Charge assigns probability x to residue class mod c iff x ∈ [0, 1/y], where y is largest prime factor of c.

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 $c$.