TITLE: Number of elements in the set $\{1,\cdots,n\}\cdot\{1,\cdots,n\}$
QUESTION [53 upvotes]: 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 $o(n^2)$?

REPLY [15 votes]: 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 $\{1,2,\dots,n^2\}$