Number Theory – Effective Upper Bound for Number of Prime Divisors

Question

Let $\omega(n) = \sum_{p \mid n} 1$. Robin proves for $n > 2$,
\begin{align}
\omega(n) < \frac{\log n}{\log \log n} + 1.4573 \frac{\log n}{(\log \log n)^{2}}.
\end{align}
Is there a similar tight effective upper bound for $\Omega(n) = \sum_{p \mid n} \text{ord}_{p}(n)$ or at least an upper bound in terms of $\omega(n)$?

Answer 1

The number of prime divisors counted with multiplicity is maximized for powers of $2$ and so

$$\Omega(n)\le\frac{\log n}{\log 2}=\log_2 n$$

and since it is exactly equal for infinitely many $n$ it is also the tighest possible bound.