Serial expansion of inverse function of $\frac{x}{\ln x}$

Question

Since $G(x)=\frac{x}{\ln x}$ is asymptotical equivalent with the prime counting function $\pi(x)$, the inverse $G^{-1}(n)$ must be asymptotical equivalent with the prime number function $p_n$.

$x=G^{-1}(n)$ can be calculated for $n\geq 3$ by $x=\underset{k\to\infty}{\lim}x_k$, where $x_{k+1}=n\cdot\ln x_k$ where $x_0=3$, but I would like to have the serial expansion anyway.

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In fact, $\; n\ln n<p_n<G^{-1}(n)\;$ and the mean $f(n)=\frac{n\ln n+G^{-1}(n)}{2}$ follows $p_n$ rather well.
Down $p_n$ is blue and $f(n)$ is red:

Answer 1

You can use the Lagrange inversion formula. But for any branch of the inverse the radius of convergence of the series will be finite, unusable for any asymptotical reasoning.