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:

Ford 2008: The distribution of integers with a divisor in a given interval. arXiv link

Koukoulopoulos 2010: Localized Factorization of Integers. arXiv link

Koukoulopoulos 2012: On the number of integers in a generalized multiplication table. arXiv link


Remark:  The dates used above refer to the publication dates (not necessarily the date posted to the arXiv).
