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.