Asymptotic Growth of Unitary Totient

Question

What is the asymptotic growth rate of the unitary totient function, $\phi^*(n)$?

It appears that $$\phi^*(n)\geq c\frac{n}{\ln n}$$ but I am sure there is a stronger lower bound.

Any linkes to references or resources are greatly appreciated.

Answer 1

It will take a few minutes for a complete proof. but I already think that Ramanujan's procedure gives exceedingly low values of this at the primorials. In which case

$$ \liminf \; \frac{e^\gamma \phi^\ast(n) \log \log n}{n} = 1. $$

Yep, it works. See both my answers at Is the Euler phi function bounded below? and my answer at Euler's Phi Function Worst Case

However, no separate proof is really necessary. Everything comes from the results of Nicolas on primorials and $\phi$ along with $$ \phi^\ast(n) \geq \phi(n). $$