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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? http://books.google.com.hk/books?id=5FlIgsWqzC8C&pg=PA157&lpg=PA157&dq=Hardy-Ramanujan+inequality&source=bl&ots=c_pxBCvkLB&sig=pReo40HZoSqARLXAd50VP3cWlwo&hl=zh-CN&sa=X&ei=KBWAUNamFOiaiQeCgYHIBg&ved=0CGoQ6AEwCA#v=onepage&q=Hardy-Ramanujan%20inequality&f=false2012-10-18
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    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!2012-10-18
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    It was sufficient. Thanks a lot :-)2012-10-24
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    Maybe add a brief answer so this doesn't appear in the unanswered queue?2013-10-14

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