$n$th prime & prime number theorem

Question

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.

gammatester · Accepted Answer · 2014-05-21T11:39:31.100

6

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:

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

edited May 21 '14 at 11:39

answered May 21 '14 at 11:29

gammatester

19,147

Great reference - thanks! – martin May 21 '14 at 11:37
Do you know if there is an English translation of Cipolla's 1902 proof of this expansion? – martin May 21 '14 at 11:43
Sorry, I do not know of such a translation. – gammatester May 21 '14 at 11:46
Many thanks anyway :) – martin May 21 '14 at 11:46
1

There is an other work from Dusart (Sharper bounds for $\psi, \theta, \pi, p_k$: http://www.unilim.fr/laco/rapports/1998/R1998_06.pdf or http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.165.6978&rep=rep1&type=pdf – gammatester May 21 '14 at 11:50
Small note - this expansion is very similar to $n\log(n\log(n))$ for small $n$. – martin May 21 '14 at 11:59

score 4 · Answer 2 · answered May 21 '14 at 11:54

4

See also The n-th prime asymptotically by Juan Arias de Reyna and Toulisse Jeremy.

answered May 21 '14 at 11:54

lhf

221,500

$n$th prime & prime number theorem

2 Answers2