Let $p\_n$ be the $n$-th prime, then from Wikipedia I got that


$p\_n \approx n \left(\ln n + \ln \ln n -1 + \frac{\ln \ln n-2}{\ln n}+\frac{6\ln \ln n-( \ln \ln n)^2-11}{\ln^2 n} \right)$.


What is a better approximation that includes 
$O\big(\frac{(\ln \ln n)^3}{\ln^3 n}\big)$? 



$n$th prime: a better approximation
You can find an in-depth answer to your question in this [paper of de Reyna and Jeremy](https://arxiv.org/abs/1203.5413). See in particular (65)-(66) along with (30) and Theorem 4.9. See also Theorem 6.2.


