# Proof of Hardy-Ramanujan inequality in number theory.

On page 3 of http://www.math.dartmouth.edu/~carlp/Lehmer0.5.pdf the author write that the following inequalities follow from "the Hardy-Ramanujan inequality", but he doesn't point to a proof. The inequalities state that $$\# \left \{ n \le t \mid \omega(n) \ge \lambda \log \log t \right \} = O \left ( \frac{e^{\lambda}t}{(\log t)^{1+\lambda \log (\lambda/e)}}\right )$$ $$\# \left \{ n \le t \mid \omega(n) \le \lambda \log \log t \right \} = O \left ( \frac{1}{(\log t)^{1+\lambda \log (\lambda/e)}}\right )$$ hold uniformly for $\lambda \ge 1$ and $0 < \lambda \le 1$ respectively, where $\omega(n)$ is the function that counts the number of different prime divisors in $n$.

Can anyone help me to find a proof?

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Is it a direct corollary of this? books.google.com.hk/… – Golbez Oct 18 '12 at 14:59
It looks very promising! I cannot say for sure before I've had some time to think about it, but that looks like that is what I want! Thanks! – Mathias Bæk Tejs Knudsen Oct 18 '12 at 15:17
It was sufficient. Thanks a lot :-) – Mathias Bæk Tejs Knudsen Oct 24 '12 at 19:06
Maybe add a brief answer so this doesn't appear in the unanswered queue? – daniel Oct 14 '13 at 12:32