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I need some tips about this:

It has been proved that

(1) $$x(\ln(x\ln(x))-1)<p_x<x\ln(x\ln(x)),\quad x>5$$

Is there a better results?

Thanks!

Arthur
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Andrea
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    Please use either \cdot or nothing at all rather than * for multiplication next time. Also note that \ln x or \ln (x) produces nicer results ($ln(x)$ versus $\ln(x)$). – Arthur Feb 19 '15 at 15:15

1 Answers1

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Pierre Dusart has improved these estimates in several papers, e.g. here, 2010. Proposition $6.6$ states that for $k\ge 688383$ we have $$ p_k\le k\left( \log k +\log_2 k -1 +\frac{\log_2 k-2.0}{\log k}\right); $$ and in Proposition $6.7$ it is stated that for all $k\ge 3$ we have $$ p_k\ge k\left( \log k +\log_2 k -1 +\frac{\log_2 k-2.1}{\log k}\right). $$ Here $\log_2 x$ means $\log(\log(x))$.

Dietrich Burde
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