Multiplication table bound from distinct prime factors
= Multiplication table bound from distinct prime factors
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>.