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 Erdős in 1955. Erdős 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 $$\left\lbrace a_1\cdots a_{k+1}\ :\ a_i\leq N \text{ for all } \ i\rbrace\right\asymp \frac{N^{k+1}}{(\log N)^{c_k}(\log\log N)^{3/2}},$$ 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). 


Let me give here an answer with a quick argument why it is $o(n^2)$. I do not know whether it is the same as Erdős original proof. UPD: it really is, and is mentioned above in a comment by Kevin P. Costello. Most numbers from 1 to $n$ have $\log \log n (1+o(1))$ prime divisors (counted with multiplicity) by ErdősKac theorem. Then most their products have $2\log \log n (1+o(1))$ prime factors, while most numbers from 1 to $n^2$ have again $\log \log n (1+o(1))$ prime factors. It proves that products of two numbers from 1 to $n$ are rare in $\{1,2,\dots,n^2\}$ 


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

