Let $\pi(m, n)$ denote the set of prime numbers in the interval $[m,n]$.
I already proved that: $$\prod_{p\in \pi(m+1, 2m)} p \leq \binom{2m}{m} $$
$$\prod_{p\in \pi(1, n)} p \leq {4^n} $$
now, I need to show that -
$${|\pi(1,n)| \leq O(n/log(n))}$$
any thoughts? would appreciate your help.