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Let $p_n$ be the $n$th prime. If $\pi(n)\sim \dfrac{n}{\log (n)}$ then $p_n\sim n\log n$ (Hardy 1938). A closer approximation is $\pi(n)\sim\text{Li}(n)$. Is there a similarly improved definition for $p_n$?

NB - I felt this was intrinsically a different question to this one, and possibly an extension of this one.

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Yes, see e.g. P. Dusart: The kth prime is greater than k(log k + log log k - 1) for k>=2, Math. Comp. 68, p 411, 1999, or online at http://functions.wolfram.com/13.03.06.0004.01:

$$p_n \approx n \left(\log(n) + \log(\log(n)) - 1 + \frac{\log(\log(n)) - 2}{\log(n)}\\ - \frac{\log(\log(n))^2 - 6 \log(\log(n)) + 11}{2 \log(n)^2} \;+ \dots\right),\quad n>2 $$

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  • $\begingroup$ Great reference - thanks! $\endgroup$ – martin May 21 '14 at 11:37
  • $\begingroup$ Do you know if there is an English translation of Cipolla's 1902 proof of this expansion? $\endgroup$ – martin May 21 '14 at 11:43
  • $\begingroup$ Sorry, I do not know of such a translation. $\endgroup$ – gammatester May 21 '14 at 11:46
  • $\begingroup$ Many thanks anyway :) $\endgroup$ – martin May 21 '14 at 11:46
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    $\begingroup$ There is an other work from Dusart (Sharper bounds for $\psi, \theta, \pi, p_k$: unilim.fr/laco/rapports/1998/R1998_06.pdf or citeseerx.ist.psu.edu/viewdoc/… $\endgroup$ – gammatester May 21 '14 at 11:50
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See also The n-th prime asymptotically by Juan Arias de Reyna and Toulisse Jeremy.

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