I am a bit stuck on how to derive the following estimate for the $n$-th prime.
$$p_n = n \log n + n \log \log n + \mathcal{O} (n)$$
(We are given that $\pi (x) = Li(x) + \mathcal{O}(\exp(-a \sqrt{\log x}))$ for some constant $a$.)
I am a bit stuck, and looking online, I just seem to find references to a very old paper by I.M. Pervushin (which I cannot actually find), and this result seems to just be assumed in a lot of literature.
Can we derive this directly from just the PNT: $\pi(x) \sim \frac{x}{\log x}$?
Any help or hints would be appreciated.