What is the tightest lower bound currently known for the Carmichael function?


I imagine it must grow much more slowly than the Euler's totient function which according to [here](https://math.stackexchange.com/questions/301837/is-the-euler-phi-function-bounded-below) is bounded as 


$$ \phi(n) \ge \frac{n}{e^{\gamma} \log \log n+ \frac{3}{\log \log n}} $$


But searching around on Google it's hard to find any literature on lower bounds of the carmichael function



Lower bound on Carmichael Function
An answer is given in Theorem 1 of the paper *[Carmichael's lambda function](https://math.dartmouth.edu/~carlp/PDF/lambda.pdf)*, by Erdos, Pomerance, and Schmutz.


**Theorem $1$**: For every $c < 1/\log{2}$, we have $\lambda(n) > (\log{n})^{c \log\log\log{n}}$ for all large enough $n$. On the other hand, for some constant $c'$, and infinitely many $n$, we also have $\lambda(n) < (\log{n})^{c'
 \log\log\log{n}}$. 


