It is well-known that $|\pi(x)-\operatorname{li}(x)| \leq \epsilon(x)$, where $\pi(x) = \sum \limits_{p \leq x} 1$ is the prime counting function, where $\operatorname{li}(x) = \int \limits_{2}^{x}\frac{1}{\ln(t)} dt$ is the logarithmic integral, and where the function $\epsilon$ satisfies $\lim \limits_{x\to\infty} \frac{\epsilon(x)}{\operatorname{li}(x)} = 0$. My question is as follows:

Is anything known about an upper bound for $|p_n - \operatorname{li}^{-1}(n)|$? Here $p_n$ is the $n$-th prime.

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