Number of elements in the set $\{1,\cdots,n\}\cdot\{1,\cdots,n\}$ 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 $o(n^2)$?  
 A: 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).
A: 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 Erdős-Kac 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 $\{1,2,\dots,n^2\}$
A: The answer is yes, for further infos see the references given at the On-Line Encyclopedia of Integer Sequences.
