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Has anyone tried to make a connection between the Mandelbrot set and the non-trivial zeros the zeta function?

Looking at the Mandelbrot set, it appears that all points are to the left of the line 0.5 + t*I, and to the right of the point -2. So the set is bounded on the left by the trivial zeros of the rienann zeta function, and appears bounded on the right by the non-trivial zeros on the critical line. Also, the set is symmetric around the real line, just like the zeta zeros. Also, the bulbs around the main cardiod involve the Farey sequence, and there is a theorem by Franel and Landau that relates the Farey sequence to the Riemann hypothesis. There are conjectures that the distribution of non-trivial zeros follows rules of quantum chaos, so potentially there could be a connection to fractals. Obviously it could all be coincidental, but it is interesting to speculate.

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    $\begingroup$ If you look hard enough, you'll find apparent connections between any two things... $\endgroup$ – Bruno Joyal Sep 14 '13 at 17:52
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The short answer is "yes". Please have a look at this link and the paper of S. C. Woon cited in the text:

http://www.dhushara.com/DarkHeart/geozeta/zetageo.htm

and for non-Mandelbrot set fractals:

http://primepatterns.tumblr.com/tagged/Riemann-zeta-function

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