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...



Asymptotic bounds on $\pi^{-1}(x)$ (inverse prime counting function)
Here is how we invert the prime counting function $\pi(x)$ to estimate the n-th prime $p\_n$. Let $g(u),u \ge 2$ be positive, continuous increasing function and let $f(x)$ be defined by


$$
x = \int\_{2}^{f(x)} \frac{du}{g(u)}
$$


Then 


$$
f(x) = \int\_{2}^{x} g(f(u))du + f(2)
$$


Take $g(u) = \ln u$ so our first estimate of $f(x) is $that $f(x) \sim x\ln x$. Now use this estimate in the above integral to obtain the second estimate for $f(x)$. You can repeat this iterative process $k$ times to obtain the asymptotic expansion of $f(x)$ with an error $O(x/(\ln x)^{-k})$.


Now from the Prime number theorem, we have


$$
\pi(x) \sim \int\_{2}^{x} \frac{du}{\ln(u)}
$$


Taking $x = p\_n$, the n-th prime, the above method says that the asymptotic expansion of $p\_n$ is $f(n)$. Thus we obtain


$$
p\_n \sim n\ln n + n\ln\ln n - n + \ldots
$$


