I hope it doesn't seem inappropriate for me to raise on MO an unanswered question from MSE, indeed a question actually posed there by someone other than myself.

I want to ask the following:

1) Consider the function $$f(z)=\sum^{\infty}_{n=1}\frac{z^{n^{2}}}{n^{2}}.$$ By the original post it is highly likely that it has fractal behavior on the circle $|z|=1$. Lacking access to Maple, I do not have the means to generate such a graph so that I may enlarge it myself to check. The commentators found the following:

A: This "fractal" behavior seems to appear in a wide range of complex functions. Alex Jordan noted this holds for *any* function of the type $\sum^{\infty}_{n=1}\frac{z^{g(n)}}{g(n)}$ where $g$ grows fast enough.

B: The imaginary part is essentially the Weierstrass function. And Riemann's "nowhere differentiable" function appears as well.

C: Slight modification (consider $f(z)=z$ [???], etc) generates similar fractal behavior.

D: $f'(z)$ seems to be essentially the well-studied Jacobi elliptic function $f(z)=\frac{1}{2}+\frac{1}{2}\theta_{3}(0,z)$.

2) I know that complex dynamics has been well studied over the past two decades, but it is not my specialty and my knowledge of the field does not suffice for understanding this problem.

I want to ask:

E: Is this something essentially new? For this function in particular, it there any association with classical elliptic functions? (I do not know much about analytical number theory.)

F: Since this function is not analytic in most of the boundary points, is there a way to describe the boundary behavior in terms of the zeros of $f'(z)$, $f''(z)$..etc?

G: Is there a way to calculate the Hausdorff dimension of it?

I do not know if this question rises to research level.