In number theory, the prime omega functions \omega(n) and \Omega(n) count the number of prime factors of a natural number n.
Thereby  \omega(n) (little omega) counts each distinct prime factor, whereas the related function \Omega(n) (big omega) counts the total number of prime factors of n, honoring their multiplicity (see arithmetic function).
That is, if we have a prime factorization of n of the form n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k} for distinct primes p_i (1 \leq i \leq k), then the respective prime omega functions are given by \omega(n) = k and \Omega(n) = \alpha_1 + \alpha_2 + \cdots + \alpha_k.
These prime factor counting functions have many important number theoretic relations.
Properties and relations
The function \omega(n) is additive and \Omega(n) is completely additive.
\omega(n)=\sum_{p\mid n} 1
If p divides n at least once we count it only once, e.g. \omega(12)=\omega(2^2 3)=2.
\Omega(n) =\sum_{p^\alpha\mid n} 1 =\sum_{p^\alpha\parallel n}\alpha
If p divides n \alpha \geq 1 times then we count the exponents, e.g. \Omega(12)=\Omega(2^2 3^1)=3.
As usual,  p^\alpha\parallel n  means \alpha  is the exact power of p dividing n .
\Omega(n) \ge \omega(n)
If  \Omega(n)=\omega(n)  then n is squarefree and related to the Möbius function by
\mu(n) = (-1)^{\omega(n)} = (-1)^{\Omega(n)}
If \Omega(n)=1  then n is a prime number.
It is known that the average order of the divisor function satisfies 2^{\omega(n)} \leq d(n) \leq 2^{\Omega(n)}.This inequality is given in Section 22.13 of Hardy and Wright.
Like many arithmetic functions there is no explicit formula for \Omega(n) or \omega(n)  but there are approximations.
An asymptotic series for the average order of \omega(n) is given by S. R. Finch, Two asymptotic series, Mathematical Constants II, Cambridge Univ. Press, pp.
21-32,
\frac{1}{n} \sum\limits_{k = 1}^n \omega(k) \sim \log\log n + B_1 + \sum_{k \geq 1} \left(\sum_{j=0}^{k-1} \frac{\gamma_j}{j!} - 1\right) \frac{(k-1)!}{(\log n)^k},
where B_1 \approx 0.26149721 is the Mertens constant and \gamma_j are the Stieltjes constants.
The function \omega(n) is related to divisor sums over the Möbius function and the divisor function including the next sums.Each of these started from the second identity in the list are cited individually on the pages Dirichlet convolutions of arithmetic functions, Menon's identity, and other formulas for Euler's totient function.
The first identity is a combination of two known divisor sums cited in Section 27.6 of the  NIST Handbook of Mathematical Functions.
\sum_{d\mid n} |\mu(d)| = 2^{\omega(n)}
\sum_{d\mid n} |\mu(d)| k^{\omega(d)} = (k+1)^{\omega(n)}
\sum_{r\mid n} 2^{\omega(r)} = d(n^2)
\sum_{r\mid n} 2^{\omega(r)} d\left(\frac{n}{r}\right) = d^2(n)
\sum_{d\mid n} (-1)^{\omega(d)} = \prod\limits_{p^{\alpha}||n} (1-\alpha)
\sum_{\stackrel{1\le k\le m}{(k,m)=1}} \gcd(k^2-1,m_1)\gcd(k^2-1,m_2) =\varphi(n)\sum_{\stackrel{d_1\mid m_1} {d_2\mid m_2}} \varphi(\gcd(d_1, d_2)) 2^{\omega(\operatorname{lcm}(d_1, d_2))},\ m_1, m_2 \text{ odd}, m = \operatorname{lcm}(m_1, m_2)
\sum_\stackrel{1\le k\le n}{\operatorname{gcd}(k,m)=1} \!
\!
\!
\!
1 = n \frac {\varphi(m)}{m} + O \left ( 2^{\omega(m)} \right )
The characteristic function of the primes can be expressed by a convolution with the  Möbius function:This is suggested as an exercise in Apostol's book.
Namely, we write f = \mu \ast \omega where f(n) = \sum_{d|n} \mu(n/d) \sum_{r|d} \left(\pi(r) - \pi(r-1)\right).
We can form the Dirichlet series over f as D_f(s) := \sum_{n \geq 1} \frac{f(n)}{n^s} = P(s), where P(s) is the prime zeta function.
Then it becomes obvious to see that f(n) = \pi(n) - \pi(n-1) = \chi_{\mathbb{P}}(n) is the indicator function of the primes.
\chi_{\mathbb{P}}(n)         = (\mu \ast \omega)(n) = \sum_{d|n} \omega(d) \mu(n/d).
A partition-related exact identity for \omega(n) is given by This identity is proved in the article by Schmidt cited on this page below.
\omega(n) = \log_2\left[\sum_{k=1}^n \sum_{j=1}^k \left(\sum_{d\mid k} \sum_{i=1}^d p(d-ji) \right) s_{n,k} \cdot |\mu(j)|\right],
where p(n) is the partition function, \mu(n) is the Möbius function, and the triangular sequence s_{n,k} is expanded by
s_{n,k} = [q^n] (q; q)_\infty \frac{q^k}{1-q^k} = s_o(n, k) - s_e(n, k),
in terms of the infinite q-Pochhammer symbol and the restricted partition functions s_{o/e}(n, k) which respectively denote the number of k's in all partitions of n into an odd (even) number of distinct parts.This triangular sequence also shows up prominently in the Lambert series factorization theorems proved by Merca and Schmidt (2017–2018) Continuation to the complex plane
A continuation of \omega(n) has been found, though it is not analytic everywhere.Z. Hoelscher & E. Palsson, Counting restricted partitions of integers into fractions: symmetry and modes of the generating function and a connection to \omega(t), The PUMP Journal of Undergraduate Research, 3 (2020), 277-307.
Note that the normalized \operatorname{sinc} function  \operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x}  is used.
\omega(z) = \log_2\left(\sum_{x=1}^{\lceil Re(z) \rceil} \operatorname{sinc} \left(\prod_{y=1}^{\lceil Re(z) \rceil+1} \left( x^2+x-yz \right) \right) \right)
Average order and summatory functions
An average order of both \omega(n) and \Omega(n) is \log\log n.
When n is prime a lower bound on the value of the function is \omega(n) = 1.
Similarly, if n is primorial then the function is as large as  \omega(n) \sim \frac{\log n}{\log\log n} on average order.
When n is a power of 2, then \Omega(n) \sim \frac{\log n}{\log 2}  .For references to each of these average order estimates see equations (3) and (18) of the MathWorld reference and Section 22.10-22.11 of Hardy and Wright.
Asymptotics for the summatory functions over \omega(n), \Omega(n), and \omega(n)^2  are respectively computed in Hardy and Wright as See Sections 22.10 and 22.11 for reference and explicit derivations of these asymptotic estimates.
Actually, the proof of the last result given in Hardy and Wright actually suggests a more general procedure for extracting asymptotic estimates of the moments \sum_{n \leq x} \omega(n)^k for any k \geq 2 by considering the summatory functions of the factorial moments of the form \sum_{n \leq x} \frac{\left[\omega(n)\right]!}{\left[\omega(n)-m\right]!} for more general cases of m \geq 2.
\begin{align}       \sum_{n \leq x} \omega(n) & = x \log\log x + B_1 x + o(x) \\       \sum_{n \leq x} \Omega(n) & = x \log\log x + B_2 x + o(x) \\       \sum_{n \leq x} \omega(n)^2 & = x (\log\log x)^2 + O(x \log\log x) \\       \sum_{n \leq x} \omega(n)^k & = x (\log\log x)^k + O(x (\log\log x)^{k-1}), k \in \mathbb{Z}^{+},    \end{align}
where B_1 \approx 0.2614972128 is the Mertens constant and the constant B_2 is defined by
B_2 = B_1 + \sum_{p\text{ prime}} \frac{1}{p(p-1)} \approx 1.0345061758.
Other sums relating the two variants of the prime omega functions include Hardy and Wright Chapter 22.11.
\sum_{n \leq x} \left\{\Omega(n) - \omega(n)\right\} = O(x),
and
\#\left\{n \leq x : \Omega(n) - \omega(n) > \sqrt{\log\log x}\right\} = O\left(\frac{x}{(\log\log x)^{1/2}}\right).
Example I: A modified summatory function
In this example we suggest a variant of the summatory functions S_{\omega}(x) := \sum_{n \leq x} \omega(n) estimated in the above results for sufficiently large x.
We then prove an asymptotic formula for the growth of this modified summatory function derived from the asymptotic estimate of S_{\omega}(x) provided in the formulas in the main subsection of this article above.N.b., this sum is suggested by work contained in an unpublished manuscript by the contributor to this page related to the growth of the Mertens function.
Hence it is not just a vacuous and/or trivial estimate obtained for the purpose of exposition here.
To be completely precise, let the odd-indexed summatory function be defined as
S_{\operatorname{odd}}(x) := \sum_{n \leq x} \omega(n) [n\text{ odd}],
where [\cdot] denotes Iverson bracket.
Then we have that
S_{\operatorname{odd}}(x) = \frac{x}{2} \log\log x + \frac{(2B_1-1)x}{4} + \left\{\frac{x}{4}\right\} - \left[x \equiv 2,3 \bmod{4}\right] + O\left(\frac{x}{\log x}\right).
The proof of this result follows by first observing that
\omega(2n) = \begin{cases}                     \omega(n) + 1, & \text{if } n \text{ is odd; } \\                      \omega(n), & \text{if } n \text{ is even,}                   \end{cases}
and then applying the asymptotic result from Hardy and Wright for the summatory function over \omega(n), denoted by S_{\omega}(x) := \sum_{n \leq x} \omega(n), in the following form:
\begin{align}  S_\omega(x) & = S_{\operatorname{odd}}(x) + \sum_{n \leq \left\lfloor\frac{x}{2}\right\rfloor} \omega(2n) \\       & = S_{\operatorname{odd}}(x) + \sum_{n \leq \left\lfloor\frac{x}{4}\right\rfloor} \left(\omega(4n) + \omega(4n+2)\right) \\       & = S_{\operatorname{odd}}(x) + \sum_{n \leq \left\lfloor\frac{x}{4}\right\rfloor} \left(\omega(2n) + \omega(2n+1) + 1\right) \\       & = S_{\operatorname{odd}}(x) + S_{\omega}\left(\left\lfloor\frac{x}{2}\right\rfloor\right) + \left\lfloor\frac{x}{4}\right\rfloor.
\end{align}
Example II: Summatory functions for so-termed factorial moments of &omega;(n)
The computations expanded in Chapter 22.11 of Hardy and Wright provide asymptotic estimates for the summatory function
\omega(n) \left\{\omega(n)-1\right\},
by estimating the product of these two component omega functions as
\omega(n) \left\{\omega(n)-1\right\} = \sum_{\stackrel{pq\mid n} {\stackrel{p \neq q}{p,q\text{ prime}}}} 1 =         \sum_{\stackrel{pq\mid n}{p,q\text{ prime}}} 1 - \sum_{\stackrel{p^2\mid n}{p\text{ prime}}} 1.
We can similarly calculate asymptotic formulas more generally for the related summatory functions over so-termed factorial moments of the function \omega(n).
Dirichlet series
A known Dirichlet series involving \omega(n) and the Riemann zeta function is given by This identity is found in Section 27.4 of the  NIST Handbook of Mathematical Functions.
\sum_{n \geq 1} \frac{2^{\omega(n)}}{n^s} = \frac{\zeta^2(s)}{\zeta(2s)},\ \Re(s) > 1.
We can also see that
\sum_{n \geq 1} \frac{z^{\omega(n)}}{n^s} = \prod_p \left(1 + \frac{z}{p^s-1}\right)^{-1}, |z| < 2, \Re(s) > 1,
\sum_{n \geq 1} \frac{z^{\Omega(n)}}{n^s} = \prod_p \left(1 + \frac{z}{p^s}\right)^{-1}, |z| < 2, \Re(s) > 1,
The function \Omega(n) is completely additive, where \omega(n) is strongly additive (additive).
Now we can prove a short lemma in the following form which implies exact formulas for the expansions of the Dirichlet series over both \omega(n) and \Omega(n):
Lemma.
Suppose that f is a strongly additive arithmetic function defined such that its values at prime powers is given by f(p^{\alpha}) := f_0(p, \alpha), i.e., f(p_1^{\alpha_1} \cdots p_k^{\alpha_k}) = f_0(p_1, \alpha_1) + \cdots + f_0(p_k, \alpha_k) for distinct primes p_i and exponents \alpha_i \geq 1.
The Dirichlet series of f is expanded by
\sum_{n \geq 1} \frac{f(n)}{n^s} = \zeta(s) \times \sum_{p\mathrm{\ prime}} (1-p^{-s}) \cdot \sum_{n \geq 1} f_0(p, n) p^{-ns},  \Re(s) > \min(1, \sigma_f).
Proof.
We can see that
\sum_{n \geq 1} \frac{u^{f(n)}}{n^s} = \prod_{p\mathrm{\ prime}} \left(1+\sum_{n \geq 1} u^{f_0(p, n)} p^{-ns}\right).
This implies that
\begin{align} \sum_{n \geq 1} \frac{f(n)}{n^s} & =       \frac{d}{du}\left[\prod_{p\mathrm{\ prime}} \left(1+\sum_{n \geq 1} u^{f_0(p, n)} p^{-ns}\right)\right] \Biggr|_{u=1}       =       \prod_{p} \left(1 + \sum_{n \geq 1} p^{-ns}\right) \times \sum_{p} \frac{\sum_{n \geq 1} f_0(p, n) p^{-ns}}{       1 + \sum_{n \geq 1} p^{-ns}} \\       & = \zeta(s) \times \sum_{p\mathrm{\ prime}} (1-p^{-s}) \cdot \sum_{n \geq 1} f_0(p, n) p^{-ns},  \end{align}
wherever the corresponding series and products are convergent.
In the last equation, we have used the Euler product representation of the Riemann zeta function.
\boxdot
The lemma implies that for \Re(s) > 1,
\begin{align}  D_{\omega}(s) & := \sum_{n \geq 1} \frac{\omega(n)}{n^s} = \zeta(s) P(s) \\       & \ = \zeta(s) \times \sum_{n \geq 1} \frac{\mu(n)}{n} \log \zeta(ns) \\  D_{\Omega}(s) & := \sum_{n \geq 1} \frac{\Omega(n)}{n^s} = \zeta(s) \times \sum_{n \geq 1} P(ns) \\       & \ = \zeta(s) \times \sum_{n \geq 1} \frac{\phi(n)}{n} \log\zeta(ns) \\ D_{\Omega\lambda}(s) & := \sum_{n \geq 1} \frac{\lambda(n) \Omega(n)}{n^s} = \zeta(s) \log \zeta(s), \end{align}
where P(s) is the prime zeta function and \lambda(n) = (-1)^{\Omega(n)} is the  Liouville lambda function.
The distribution of the difference of prime omega functions
The distribution of the distinct integer values of the differences \Omega(n) - \omega(n) is regular in comparison with the semi-random properties of the component functions.
For k \geq 0, let the sets
N_k(x) := \#\{n \leq x: \Omega(n) - \omega(n) = k\}.
These sets have a corresponding sequence of limiting densities d_k such that for x \geq 2
N_k(x) = d_k \cdot x + O\left(\left(\frac{3}{4}\right)^k \sqrt{x} (\log x)^{\frac{4}{3}}\right).
These densities are generated by the prime products
\sum_{k \geq 0} d_k \cdot z^k = \prod_p \left(1 - \frac{1}{p}\right) \left(1 + \frac{1}{p-z}\right).
With the absolute constant \hat{c} := \frac{1}{4} \times \prod_{p > 2} \left(1 - \frac{1}{(p-1)^2}\right)^{-1},  the densities d_k satisfy
d_k = \hat{c} \cdot 2^{-k} + O(5^{-k}).
Compare to the definition of the prime products defined in the last section of  in relation to the Erdős–Kac theorem.
See also
Additive function
Arithmetic function
Erdős–Kac theorem
Omega function (disambiguation)
Prime number
Square-free integer
Notes
References
External links
OEIS Wiki for related sequence numbers and tables
OEIS Wiki on Prime Factors
