TITLE: Ramanujan's First Letter to Hardy and the Number of $3$-Smooth Integers
QUESTION [30 upvotes]: A positive integer is $B$-smooth if and only if all of its prime divisors are less than or equal to a positive real $B$. For example, the $3$-smooth integers are of the form $2^{a} 3^{b}$ with non-negative exponents $a$ and $b$, and those integers less than or equal to $20$ are $\{1,2,3,4,6,8,9,12,16,18\}$.
In Ramanujan's first letter to G. H. Hardy, Ramanujan emphatically quotes (without proof) his result on the number of $3$-smooth integers less than or equal to $N > 1$,
\begin{eqnarray}
\frac{\log 2 N \ \log 3N}{2 \log 2 \ \log 3}.
\end{eqnarray}
This is an amazingly accurate approximation, as it differs from the exact value by less than 3 for the first $2^{1000} \approx 1.07 \times 10^{301}$ integers, as shown by Pillai. 
Question: Knowing full well that Ramanujan only gave proofs of his own claims while working in England, I wonder if a proof of this particular estimate appears somewhere in the literature. Is this problem still open? If not, what is a reference discussing its proof?
Thanks!

REPLY [16 votes]: For any reference requirement related to Ramanujan, it is always a good idea to check the series of volumes, titled Ramanujan's notebooks, compiled and annotated by Bruce C. Berndt and others.
This is a special case of Entry 15 in Ramanujan's second notebook, which is about numbers of the form $\displaystyle a^p b^q$.
A reference to this can be found here: Ramanujan's notebooks, Volume 4.
(I suggest you read page 66 onwards)
A snapshot (from the google books link itself):

This volume talks about the "proof" Ramanujan gave (pages 68 and 69), provides references to Hardy's book (which apparently has a whole chapter on this) and also mentions a paper by Hardy and Littlewood which deals with it.
De Bruijn has considered the number of $y$-smooth numbers in this paper: On the number of positive integer less than x and free of prime factors greater than y. 
The number of $y$-smooth numbers $\le x$ is apparently now known in literature as the DeBruijn function: $\psi(x,y)$. 
A closely related function is the DeBruijn-Dickman function.
There is also a survey by Hildebrand and Tenenbaum which should be helpful.