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.

  • 1
    $\begingroup$ Please, be more precise. Fractal is not a precise term. Do you mean that the set of zeros, or more likely, the set of poles on the unit circle, is a cantor set, with a positive Hausdorff dimension? I believe your situation is similar to sets like $e^\{i\theta}, \theta = \sqrt{2}\pi n, n \in \mathbb{Z}$ or the complement. $\endgroup$ – Per Alexandersson Dec 13 '11 at 9:25
  • $\begingroup$ I think your best hope is your point B. The real part of $f(e^i\theta)$ is the example $\sum \cos(n^2\theta)/n^2$ that HAS been studied, and the imaginary part with $\sin$ is similar. $\endgroup$ – Gerald Edgar Dec 13 '11 at 13:05
  • $\begingroup$ @Paxinum: I mean the image of $f(z)$ mapping the unit circle has positive Hausdorff dimension and may be self-similar. As I commented I cannot enlarge the image in the original website to check it myself, nor do I know how to verify this theoretically (otherwise I would not ask it in here). You mean the function $$e^\{i\theta}, \theta = \sqrt{2}\pi n, n \in \mathbb{Z}$$? Yes, I think so. $\endgroup$ – Kerry Dec 13 '11 at 18:44
  • $\begingroup$ @Gerald Edgar: This is not my point, it is really other people's point I saw in MSE I copied in here. Thanks. $\endgroup$ – Kerry Dec 13 '11 at 18:54
  • $\begingroup$ Edited the title as the first commentor suggested. Also the problem has been addressed in at least a few academic papers so probably no longer appropriate in here. $\endgroup$ – Kerry Feb 26 '12 at 6:04

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