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 $o(n^2)$? 



Number of elements in the set $\{1,\cdots,n\}\cdot\{1,\cdots,n\}$
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](http://arxiv.org/abs/math/0401223)
* Koukoulopoulos 2010: Localized Factorization of Integers. [arXiv link](http://arxiv.org/abs/0809.1072)
* Koukoulopoulos 2012: On the number of integers in a generalized multiplication table. [arXiv link](http://arxiv.org/abs/1102.3236)


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


