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!