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I recently read a very good inequality concerning the no of primes $\pi(x)$:

$$\pi(n)>\frac{1}{6}\frac{n}{\log n}\mathrm{\ for\ }n\ge 2$$

Are any other such elementary inequalities concerning the primes?

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    Could you just chop those blanks in page 2?2010-08-10
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    http://i33.tinypic.com/s1q6jd.png2010-08-10
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    See the discussion at http://mathoverflow.net/questions/26342/chebyshevs-approach-to-the-distribution-of-primes2010-08-10
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    @Chandru1: I've edited the question to remove the scans of the text, since you didn't give any indication of where it came from (with a citation, that kind of quoting might be okay), and to change the question to ask what you said you wanted to know in your comment on lhf's answer. If you don't agree with my edits, please feel free to re-edit the question.2010-08-10
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    This is a valid question2010-08-14

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What is your question? This is well-known stuff. It appears for instance in section 4.5 of Apostol's Introduction to Analytic Number Theory, which also contains an upper bound. You could at least say where you got yours from. (This should have been a comment, not an answer, but I don't have enough reputation for adding comments.)

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    Yes it does appears in Tom Apostol's book, but what i wanted to know is whether there are any other such elementary inequalities concerning the primes.2010-08-10