Asymptotic behavior of $\Gamma^{-1}(x)$

Solution 1:

I got a better approximation by fiddling with Stirling's formula out of Abramowitz and Stegun, just using the first term $$ \log \Gamma(z) \sim z \log z, $$ take $t = \Gamma(z)$ and the approximation $$ z_1 = \frac{L_1}{L_2} + \frac{L_1 L_3}{L_2^2}, $$ where $L_1 = \log t, \; \; L_2 = \log \log t, \; \; L_3 = \log \log \log t.$ I get $$ z_1 \log z_1 = \log t - \frac{L_1 L_3^2}{L_2^2} + \frac{L_1 L_3}{L_2^2} + smaller $$ which is an improvement on your folklore result, as the unwanted terms are actually smaller than the next term $z$ in the fuller $$ \log \Gamma(z) \sim z \log z - z - \frac{1}{2} \log z + \frac{1}{2} \log {2 \pi} + \log \left(1 + \frac{1}{12 z} + \frac{1}{288 z^2}- \cdots \right) $$ You ought to be able to do something with this for your original question.

In particular, it says your $$ G(t) \sim 1 + \frac{\log \log \log t}{\log \log t}, $$ meaning change is so slow that the limit is invisible to a computer, your experimental bounds may well be correct but certainty will be hard to come by.