# What is the asymptotic behavior of the function counting the number of (not necessarily distinct) prime divisors?

Ω(n), ω(n), νp(n) – prime power decomposition

The fundamental theorem of arithmetic states that any positive $n = p_1^{a_1}\cdots p_k^{a_k}$ where $p_1 < p_2 < ... < p_k$ are primes and the $a_j$ are positive integers. (1 is given by the empty product.)

It is often convenient to write this as an infinite product over all the primes, where all but a finite number have a zero exponent. Define $νp(n)$ as the exponent of the highest power of the prime $p$ that divides $n$. I.e. if $p$ is one of the $p_i$ then $νp(n) = a_i$, otherwise it is zero. Then

$$n=\prod_p p^{\nu_p(n)}.$$

In terms of the above the functions ω and Ω are defined by

ω(n) = k,
Ω(n) = a_1 + a_2 + ... + a_k.


To avoid repetition, whenever possible formulas for the functions listed in this article are given in terms of n and the corresponding pi, ai, ω, and Ω.

It seems that a few things are clearly known about $\omega(n)$. For instance, it has normal order $log \ log \ (n)$.

How can we bound $\Omega (n)$ in the asymptotic limit?

$\Omega[N]$ asymptotically approaches $\ln \ ln \ N$.