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# AquestionrelatedtofractalbehaviorSelf-similarity of ellipticfunctionsRiemann's"non-differentiable"function

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I do not know if hope it is appropriate doesn't seem inappropriate for me to ask raise on MO an unanswered question with no answer I saw in from MSEto MO, especially I am not the person come up with the problemindeed 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 $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 know how have the means to generate such a graph so that I may enlarge it myself to check(because I do not have Maple)check. The commenters commentators found the following: A: This "fractal" behavior seems to appear in a wide range of complex functions. Alex Jordan noted that this holds for any function of the type$\sum^{\infty}_{n=1}\frac{z^{f(n)}}{f(n)}$\sum^{\infty}_{n=1}\frac{z^{g(n)}}{g(n)}$ where $f$ g$grows fast enough. B: The imaginary part is essentially the Weistrass Weierstrass function. And Riemann's "nowhere differentiable" function appeared appears as well. C: Slight modification (consider$f(z)=z$, f(z)=z$ [???], etc) generate 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) So I know that complex dynamics has been well studied in over the past two decades, but it is not my speciality specialty and my knowledge in this of the field is does not helpful to understand suffice for understanding this problem.

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?