I am trying to compare asymptotic runtime bounds of a few algorithms presented in this research paper, A quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic. The functions are,
$L(\alpha) = \exp(O(\log(n)^{a}(\log\log(n))^{1-\alpha}))$,
which I'd like to plot for $\alpha = 1/3, 1/4+O(1)$, and $1/4 + O(n)$,
and $O(n^{\log n})$
Where n is the bit-size of the input. ... Such a complexity is smaller than any $L(\epsilon)$ for any $\epsilon > 0$
I would like to plot each of these functions together to compare their growth. The problem is that the second function grows much faster than the first, which implies I am misinterpreting something.
So what is the correct way to interpret and compare these functions?