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I have this question in mind since the first time I saw a graphical representation of the zeta function (like in the sample below). Just by looking to them I wondered if there is any relationship between the Riemann zeta function as showed in the sample and strange attractors? Disclaimer: my question is just based in the visual look of both structures.

For instance the Lorenz's strange attractor looks like this (sorry I do not know how to reduce the size of the images):

enter image description here

And a graphic of the zeta function for ($\frac{1}{2}+it$) looks like this:

enter image description here

I have been trying to find any reference regarding Riemann zeta function and attractors but if I am not wrong there is nothing and probably this is just a visual similarity but without any real background behind. Thank you!

UPDATE 2015/05/26

So far I have found some papers and some hints, but nothing clear about this yet! Here is the list:

  1. "Magic Angle Precession and the Riemann Zeta Function" (Binder): The Riemann zeta function and the magic angle precession strange attractor are related by their functional equations.

  2. Recurrent Anholonomy in Curved Space Navigation Solved by the Riemann Zeta Function (Binder).

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    $\begingroup$ Concerning iterations and fractals you may enjoy this (probably not an answer to your question but neat!) $\endgroup$ – Raymond Manzoni May 17 '15 at 18:10
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    $\begingroup$ @Raymond Manzoni, thank you for the reference! it is really interesting, I had a look to it, I guess I will need a weekend to read it slowly. :) $\endgroup$ – iadvd May 18 '15 at 0:02
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    $\begingroup$ Two more things you may enjoy : the universality of zeta and the related paper by Woon "Riemann zeta function is a fractal". Fine reading, $\endgroup$ – Raymond Manzoni May 22 '15 at 21:48
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There is no known relation. But there exist so called "dynamical zeta functions". I have newer seen anybody drawing plots like yours for them but maybe they look similar.

Ref: Lagarias, about Number Theory Zeta Functions and Dynamical Zeta Functions (pdf)

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  • $\begingroup$ thanks for the reference. I have added to your explanation (hope will be updated in some minutes) a link to: Number Theory Zeta Functions and Dynamical Zeta Functions, math.lsa.umich.edu/~lagarias/doc/numberthzeta.pdf (Lagarias) $\endgroup$ – iadvd May 17 '15 at 8:51

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