TITLE: Asymptotic bounds on $\pi^{-1}(x)$ (inverse prime counting function)
QUESTION [8 upvotes]: What are the current best asymptotic bounds on $\pi^{-1}(x)$, where $\pi(x)$ denotes the prime counting function (number of primes at most $x$)?
In other words, I am curious about the state of the art for estimating the $n^{th}$ prime. From the prime number theorem, it seems clear that $\pi^{-1}(x)=\Theta(x \log x)$. Can someone point me in the direction of literature that answers the question? Please excuse my inability to find such literature myself...

REPLY [4 votes]: Since "the state of the art" slightly changed since 2013, I am adding this new result on lower and upper bounds on $p_n$. (This is complementary to the answers given earlier.)
Christian Axler proved in https://arxiv.org/abs/1706.03651 that:
(1) For every $n\ge46254381$,
$${\small
p_n < n \Big(\log n + \log\log n - 1 + {\log\log n - 2\over\log n} - {(\log\log n)^2 - 6\log\log n + 10.667 \over 2\log^2 n} \Big).
}
$$
(2) For every $n\ge2$,
$${\small
p_n > n \Big(\log n + \log\log n - 1 + {\log\log n - 2\over\log n} - {(\log\log n)^2 - 6\log\log n + 11.508 \over 2\log^2 n} \Big).
}
$$