TITLE: Generalized PNT in limit as numbers get large
QUESTION [9 upvotes]: If $\pi_k(n)$ is the cardinality of numbers with k prime factors (repetitions included) less than or equal n, the generalized Prime Number Theorem (GPNT) is:
$$\pi_k(n)\sim \frac{n}{\ln n} \frac{(\ln \ln n)^{k-1}}{(k-1)!}.$$
The qualitative appearance of the actual distribution of $\pi_k(n)$ for k = 1,2,3,..., agrees very well with the GPNT, for numbers $n$ within reach of my laptop. But I noticed that as $n$ and $k$ get large, "most" of the numbers less than $n$ seem to have relatively few factors.   
Writing $n = 2^m$ and replacing $k$ by $x$ we can graph 
$$f(x) =\frac{2^m (\ln\ln 2^m)^{x-1}}{\ln 2^m (x-1)!}$$  
from $x = 1$ to $m$ (since no number will have more than m factors) and see that for relatively small fixed $m$, most of the area under the curve f is is contained in a steep bell-shaped curve on the far left of the image. 
I take this to suggest that as we consider very large sets, $S_m = \{ 1,2,3,...,2^m\},$ almost all elements of these sets have a "very small" number of factors (including repetitions).
Can this idea be (or has it been) quantified? The phrase "very small" is frustrating, and I think we might be able to say something more concrete about, for example, concentration of the proportion of area as a function of x and m...?  Thanks for any suggestions. 
Edit: the answer Eric Naslund gave below is splendid and I won't neglect to accept it. In response to the answer, I wonder if there is any reason not to be able to get something like that answer from the expression $f(x)$? 
After all, $f(x)$ appears to be a Poisson-like curve with a mean near the average number of prime factors. If I let m = 100 and then 500 (i.e., we're using $2^{100},2^{500}$), $f'(x) = 0$ at $x \approx 4.73, 6.34$, respectively, while $\ln\ln 2^m$ is respectively 4.23, 5.84. If f is a valid expression for the asymptotic behavior of $\pi_k(n)$, wouldn't we expect it to give us this additional information? Can we not prove it?

REPLY [5 votes]: I am not quite sure that I understand exactly what you're asking, but I think that some of the following points may shed some light on the matter.
Suppose we fix $n$ and let $k$ vary.  Studying $\pi_k(n)/n$ is then equivalent to looking at the distribution of the number of prime factors of integers up to $n$ (by the way, this turns out not to be very sensitive to the distinction between "with repetition" and "without repetition").
We have to be a bit careful as the asymptotic for $\pi_k(n)$ is intended for $k$ fixed and $n$ growing (but we can still use it as long as $k$ grows sufficiently slowly with $n$).  With that caveat in mind, the approximate distribution given by GPNT can be rewritten as:
$$\frac{\pi_{k+1}(n)}{n} \approx \frac{\lambda^{k} e^{-\lambda}}{k!},\quad k=0,1,2,\ldots$$
where $\lambda = \log \log x$, and I have shifted the value of $k$ by 1.  By rewriting it this way, we can recognize the right hand side exactly as a Poisson distribution with parameter $\lambda = \log \log x$.
Loosely interpreted, every number (above 1) is guaranteed to have at least one prime factor, but the additional primes beyond that behave Poissonly with a rate of $\log \log x$.  The peak of any Poisson distribution always occurs at $\lambda$.  You don't need to take derivatives to see this, just look at the ratio  of the $k-1$ and $k$ terms:
$$\frac{\lambda^k e^{-\lambda}}{k!} \cdot \frac{(k-1)!}{\lambda^{k-1} e^{-\lambda}}  = \frac{\lambda}{k}.$$
So clearly the Poisson distribution is increasing in $k$ for $k < \lambda$ and decreasing for $k > \lambda$, and the peak occurs at $\lfloor\lambda\rfloor$.
Moreover, it is well-known (see the linked Wikipedia article) that the normal distribution with equal mean and variance $\lambda$ approximates the Poisson distribution when $\lambda$ is large.  Perhaps this explains the bell curve you observed.
Finally, note that equal mean and variance $\log \log x$ is precisely the situation that arises from the Erdős-Kac theorem in Eric Naslund's answer, although let me reiterate that I am deliberately ignoring the relative sizes of $n$ and $k$ for which these separate facts are known to be true.