Timeline for Self-similarity of Riemann's "non-differentiable" function
Current License: CC BY-SA 3.0
14 events
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
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| Aug 4, 2012 at 9:33 | comment | added | Alexandre Eremenko | 1. Connection with elliptic functions: $$zf'(z)=\sum_{n>0}z^{n^2}.$$ Putting $z=\exp(\pi i\tau)$ we obtain $(\theta(0,\tau)-1)/2$, where $\theta$ is the standard theta-function, see Mumford, Tata lectures on theta, page 1. 2. To the best of my knowledge, the Hausdorff dimension of the Weierstrass curve is not known. | |
| Feb 26, 2012 at 6:04 | comment | added | Kerry | 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. | |
| Feb 26, 2012 at 6:02 | history | edited | Kerry | CC BY-SA 3.0 |
edited title; edited title
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| Dec 25, 2011 at 21:08 | history | edited | Goldstern |
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| Dec 13, 2011 at 18:54 | comment | added | Kerry | @Gerald Edgar: This is not my point, it is really other people's point I saw in MSE I copied in here. Thanks. | |
| Dec 13, 2011 at 18:44 | comment | added | Kerry | @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. | |
| Dec 13, 2011 at 13:05 | comment | added | Gerald Edgar | 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. | |
| Dec 13, 2011 at 9:25 | comment | added | Per Alexandersson | 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. | |
| Dec 13, 2011 at 6:56 | history | edited | David Feldman | CC BY-SA 3.0 |
Repaired grammar and English style, hopefully without changing the meaning.; added 2 characters in body
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| Dec 13, 2011 at 4:07 | history | edited | Kerry | CC BY-SA 3.0 |
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| Dec 13, 2011 at 4:03 | history | edited | Yemon Choi |
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| Dec 13, 2011 at 3:31 | history | edited | Kerry | CC BY-SA 3.0 |
deleted 2 characters in body; edited tags; edited title
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| Dec 13, 2011 at 3:05 | history | asked | Kerry | CC BY-SA 3.0 |