TITLE: Asymptotic estimation of $A_n$
QUESTION [10 upvotes]: Let $A_n$ represent the number of integers that can be written as the product of two element of $[[1,n]]$.
I am looking for an asymptotic estimation of $A_n$.
First, I think it’s a good start to look at the exponent $\alpha$ such that :
$$A_n = o(n^\alpha)$$
I think we have : $2 \leq \alpha $. To prove this lower bound we use the fact that the number of primer numbers $\leq n$ is about $ \frac{n}{\log n}$.
Hence we have the trivial lower bound (assuming $n$ is big enough) :
$$ \binom{ E(\frac{n}{\log n})}{2} = o(n^2)$$
Now is it possible to get a good asymptotic for $A_n$ and not just this lower bound ? 
Is what I’ve done so far correct ? 
Thank you !

REPLY [2 votes]: The exact asymptotic isn’t known, but Kevin Ford astutely worked out the exact order of magnitude, not too long ago.  So the only unknown is the constant in front (or perhaps even whether such a constant exists).
From the answer at https://math.stackexchange.com/a/1336590/30402, the value of $A_N$ satisfies upper and lower bounds of the form
$$
A_N \asymp \frac{N^2}{(\log N)^c(\log\log N)^{3/2}}$$ where $$c=1-\frac{(1+\log \log 2)}{\log 2}.
$$