# A tale of two omegas

## TL;DR

This paper explores the properties and differences of the prime factor counting functions (n) and (n), analyzing their summatory functions and their connections to the Riemann hypothesis, revealing distinct behaviors and oscillations.

## Contribution

It provides new insights into the oscillatory behavior of the summatory functions of (n) and (n), including bounds and bias analysis, with implications for the Riemann hypothesis.

## Key findings

- (n) and (n) are even about 73.5% of the time
- The summatory function L(x) is biased toward negative values
- H(x) exceeds 1.7or infinitely many x and is less than -1.7or infinitely many x

## Abstract

We consider $\omega(n)$ and $\Omega(n)$, which respectively count the number of distinct and total prime factors of $n$. We survey a number of similarities and differences between these two functions, and study the summatory functions $L(x)=\sum_{n\leq x} (-1)^{\Omega(n)}$ and $H(x)=\sum_{n\leq x} (-1)^{\omega(n)}$ in particular. Questions about oscillations in both of these functions are connected to the Riemann hypothesis and other questions concerning the Riemann zeta function. We show that even though $\omega(n)$ and $\Omega(n)$ have the same parity approximately 73.5\% of the time, these summatory functions exhibit quite different behaviors: $L(x)$ is biased toward negative values, while $H(x)$ is unbiased. We also prove that $H(x)>1.7\sqrt{x}$ for infinitely many integers $x$, and $H(x)<-1.7\sqrt{x}$ infinitely often as well. These statements complement results on oscillations for $L(x)$.