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For real $x,$ it's well-known that $$\Gamma^{-1}(x)\sim\frac{\log x}{\log\log x}$$

So a natural question is to bound $$G(x)=\Gamma^{-1}(x)\frac{\log\log x}{\log x}$$ which of course is 1 + o(1). Interestingly, its value is near 2 (that is, far from its asymptotic value) for many useful values of x: for example, $1.8<G(x)<2.1$ for $14<x<10^{77}.$

It seems that $G$ has a maximum near 3637.133905003816072848664328 of 2.01119450670696919822787997170113557148977275... and to decrease very slowly thereafter.

Question 1: Is the above the unique maximum?

Question 2: Is there an $x>5$ such that $G(x)<1$?

Question 3: Is there a useful factor or secondary term that makes this approximation more precise for useful values of x? I'm being intentionally vague on this point—if I knew exactly what I was looking for I probably wouldn't need to ask. :) For example, had I asked an analogous question about the prime-counting function, telling me about Li would be better than just giving the next asymptotic term $cx/\log^k x.$

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I don't have the time to look now, but here's an idea: how about studying the graph of $\arctan(G(\tan(x)))$ within $(0,\pi/2)$? –  J. M. May 3 '11 at 17:50
Can you provide a reference for your "well-known" fact? –  GEdgar May 3 '11 at 19:09
Perhaps study $G(\Gamma(y)) = y \log\log\Gamma(y)/\log\Gamma(y)$. –  GEdgar May 3 '11 at 19:15
@GEdgar: I can give sources using it, but I don't know of any addressing it directly -- it's folklore. –  Charles May 3 '11 at 20:12
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up vote 2 down vote accepted

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.

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