Let $A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $A_n$? Will it be $o(n^2)$?

This question is known as the multiplication table problem, and was originally posed by Erdos in 1955. Erdos proved that $A_n=o(n^2)$, and this was sharpened by Tenenbaum in 1984. In 2008, Ford gave the exact magnitude and proved that $$\left\lbrace a\cdot b:\ a,b\leq N\rbrace\right\asymp \frac{N^2}{(\log N)^c(\log\log N)^{3/2}}$$ where $$c=1\frac{(1+\log \log 2)}{\log 2}.$$ In 2010 Koukoulopoulos gave multidimensional generalizations of Ford's result, proving that $$\left\lbrace a_1\cdots a_k\ :\ a_i\leq N \text{ for all } \ i\rbrace\right\asymp \frac{N^{k+1}}{(\log N)^{c_k}(\log\log N)^{3/2}}$$ where $$c_{k}=\int_{1}^{\frac{k}{\log(k+1)}}\log x\text{d}x=\frac{\log(k+1)+k\log\left(k\right)k\log\log(k+1)k}{\log(k+1)}.$$ Some references:
Remark: The dates used above refer to the publication dates. (Not necessarily the date posted to the arXiv) 


The answer is yes, for further infos see the references given at the OnLine Encyclopedia of Integer Sequences. 

