In the paper Picture of an essential singularity, an analogy is made between the multipolar moments of infinitesimal charge distributions and the lines of constant modulus/argument around an essential singularity. Further, the usual residue formula
$$ \frac{1}{2\pi i}\int_C z^n \frac{f'(z)}{f(z)} dz $$
is used to find the $n$-th moment by integrating around the essential singularity. **My question is if these moments uniquely define the behavior of an analytic function around the essential singularity.** And if not, is there a way to characterize essential singularities in the same way that we do with residues for poles?

I wonder this because a symmeterized and inverted function of the Riemann zeta function $$ \zeta\left(w\right)\Gamma\left(w/2\right)\pi^{-w/2}w(1-w) $$ where $w=1/z+1/2$ has only an essential singularity at the origin, and is otherwise well behaved. If the Riemann hypothesis is true, I wonder if it is equivalent to a statement along the lines of: the essential singularity has no moments higher than a quadrupole.