I'm reading the paragraph 2.3 about the Replica Method and I fail to see how the first 3 formulas are derived one from another.
Given the definition of
$$ f = - \frac 1 {\beta N} \mathbb E \ln Z$$
and using the identity
$$ \lim_ {n \to 0} \frac {Z^n - 1} n = \ln Z $$
how can they immediately derive the following?
$$ f = \lim_ {n \to 0} - \frac 1 {n \beta N} \ln \mathbb E Z^n$$
While it is clear to see that a factor $\frac 1 n$ is introduced together with the replica $Z^n$ (and I could guess that - 1 is maybe dropped because it will vanish in the thermodynamic limit?), I'm wrapping my head around the $\mathbb E$ inside a $\ln$: what are the steps to this equation?
At first sight, I've suspected it is just a typo, but, for example, a youtube video of Francesco Guerra at minute 14:07 is actually showing a very similar auxiliary function with $\mathbb E Z^s_N(\beta)$ inside a log, hence I'm not really sure if it is a typo or something deeper is going on here.