Residue races of the number of prime divisors function

TL;DR

This paper studies how the number of distinct prime divisors of integers distributes across different residue classes modulo q, exploring prime number race phenomena for various moduli.

Contribution

It extends the analysis of the distribution of ω(n) in residue classes to larger moduli, addressing prime race questions inspired by prior conjectures.

Findings

Distribution patterns of ω(n) in residue classes identified

Prime number race behaviors analyzed for various moduli

Insights into biases in prime divisor counts across classes

Abstract

We investigate the distribution of the function $\omega(n)$, the number of distinct prime divisors of $n$, in residue classes modulo $q$ for natural numbers $q$ greater than 2. In particular we ask `prime number races' style questions, as suggested by Coons and Dahmen in their paper `On the residue class distribution of the number of prime divisors of an integer'.