How do we prove $p_n\sim n\log(n\log(n))$ from the Prime Number Theorem?

Question

Let $p_n$ be the $n$th prime.

Could someone please help me with the steps between $\pi(n)\sim\dfrac{n}{\log(n)}$ and $n=\pi(p_n)$, to the statement $p_n\sim n\log(n\log(n))$?

Answer 1

Here is a sketchy derivation.

Note that

$$ n \log(n \log n) = n \log n + n \log \log n \sim n \log n $$

so we just need to prove the simpler statement that $p_n \sim n \log n$.

The formulas

$$ \pi(n) \sim \frac{n}{\log n} $$ $$ \pi(p_n) \sim n $$

imply

$$ \begin{align} n &\sim \frac{p_n}{\log p_n} \\ p_n &\sim n \log p_n \\ &\sim n \log(n \log p_n) \\ &\sim n (\log n + \log \log p_n) \\ &\sim n \log n \end{align}$$