In the Math World article on Merten's Constant, a related constant $B_2$ is mentioned which "appears in the summatory function of the number of (not necessarily distinct) prime factors."
I am very unclear what this means. I clicked on the link for summatory function and I find the definition of the summatory function for $f(n)$ unclear:
$$F(n) \equiv \sum_{k\in D} f(k)$$
where $D$ is the domain of the function.
It would be very helpful if someone could explain how this equation:
$$\sum_{n\le x}\Omega(n)=x\ln\ln x +B_2x + o(x)$$
relates to the number of not necessarily distinct prime factors.
Am I right to assume that the article has a typo and it should be $\Omega(x)$ instead of $\Omega(n)$?
I tried to apply this equation to $100=2^25^2$ which has $4$ prime factors.
$\Omega(100) = 100\ln\ln 100 + 1.034653... = 152.7179626...$ which shows that I am misunderstanding the equation.
Could someone explain to me what the number of not necessarily distinct prime factors means and show how it relates to an example such as $100$?
The function $\Omega(n)$ is, by definition, the number of prime factors of $n$, counted with multiplicity (that is, not necessarily distinct). For example, $\Omega(720) = \Omega(2^43^25) = 4+2+1 = 7$.
The summatory function for $\Omega$ is indeed $F(x) = \sum_{n\le x} \Omega(n)$. For example, $F(8) = \Omega(1) + \Omega(2) + \cdots + \Omega(8) = 0 + 1 + 1 + 2 + 1 + 2 + 1 + 3 = 11$. (And $F(8.7)$ is also $11$.)
The theorem you're quoting says that $\sum_{n\le x} \Omega(n) = x\ln\ln x + B_2x + o(x)$. In other words, $$ \lim_{x\to\infty} \frac{F(x) - x\ln\ln x}x = B_2. $$