I saw a proof that $|p\_n - li^{-1}(n)| \leq n e^{-c \sqrt{\ln(n)}} $, 
without saying anything about $c$ !


My questions is, what the explicit value of $c$ ??


It just says for some number $c$ without giving any information about $c$ !



An explicit value for a bound proof
An explicit version of the Prime Number Theorem required here is Theorem 1.12 on Page 42 of [Dusart's thesis](http://www.unilim.fr/laco/theses/1998/T1998_01.pdf). In particular, it follows from this theorem that
$$ |\pi(x)-\mathrm{Li}(x)|<x e^{-0.32\sqrt{\ln x}},\qquad x\geq 59. $$
Using Kadiri's explicit zero-free region, [Trudgian](https://arxiv.org/abs/1401.2689) sharpened this result to
$$ |\pi(x)-\mathrm{Li}(x)|<x e^{-0.39\sqrt{\ln x}},\qquad x\geq 229.$$


Strictly speaking, these inequalities do not answer the original question. On the other hand, applying either of them at $x=p\_n$, one obtains a similar bound for $|n-\mathrm{Li}(p\_n)|$ in terms of $n$ (using a crude but concrete Chebyshev type bound $p\_n\asymp n\log n$), and then applying $\mathrm{Li}^{-1}$ to both $n$ and $\mathrm{Li}(p\_n)$, one concludes an analogous bound for $|\mathrm{Li}^{-1}(n)-p\_n|$.


**Added.** See [arXiv:2206.12557](https://arxiv.org/abs/2206.12557) and the references therein for even sharper results.


