What is the largest constant $c$ such that $$ \frac{c\,x}{\ln x} < \pi(x)$$ for all integers $x$? Also, if no one knows, do you think expecting an answer is unreasonable?
Prime counting inequality
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number-theory
inequality
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1For all integers $n$? What's $n$? If $n=x$, what if $x=1$? – 2012-11-14
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0Not quite what you want, but see http://math.stackexchange.com/a/59270/589. – 2012-11-14
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2Expecting an answer is unreasonable. – 2012-11-14
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1I think the answer is $\pi(2)/(2/\ln(2))=\ln(2)/2$ (but you'll have to live with $\le$ instead of $<$. – 2012-11-14
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0@WillJagy Coming up with a lower bound as above is actually not hard, and lhf has the correct answer (though without a proof). I have detailed a proof of this bound in my answer, below. – 2012-11-15