Multiplication table problem ($M(N)=\#\{ab:1\le a,b\le N\}$)

The multiplication table problem asks how many distinct integers occur as $ab$ with $1\le a,b\le N$. Counting the $N^2$ pairs greatly overcounts the distinct products.

Concentration of the prime omega function near $\log \log N$, together with $\omega (ab)=\omega (a)+\omega (b)-\omega (\gcd (a,b))$ and the bounded mean of $\omega (\gcd (a,b))$, gives $M(N)\ll N^2/\log \log N$. This elementary argument uses only distinct prime factors.