Print Page - Riemann zeta function as fractal how ?

Title: Riemann zeta function as fractal how ?
Post by: flexiverse on March 05, 2015, 05:13:14 AM

If the Riemann zeta function is actually a fractal.......

What fractal is it? Any programs or code that shows it:

See this paper: http://xxx.lanl.gov/pdf/chao-dyn/9406003v1

Title: Re: Riemann zeta function as fractal how ?
Post by: DarkBeam on March 05, 2015, 09:07:46 AM

Riemann Z is ultra complicated. :)

Here a discussion about it: http://math.stackexchange.com/questions/183680/modern-formula-for-calculating-riemann-zeta-function

A video by an italian guy ;)
http://www.youtube.com/watch?v=QFuGqx4ntKY

Imho... fractals should use simple functions. Those overly strange rarely give good results

Title: Re: Riemann zeta function as fractal how ?
Post by: laser blaster on March 05, 2015, 06:28:06 PM

If that paper is correct, the function itself is a fractal, however it's not a very pretty fractal by itself. However, the zeta function has it's own mandelbrot and julia sets, like any function on the complex plane. Some people have visualized them and studied them here:

http://www.dhushara.com/DarkHeart/geozeta/zetageo.htm (http://www.dhushara.com/DarkHeart/geozeta/zetageo.htm)
http://www.dhushara.com/DarkHeart/RH2/RH.htm (http://www.dhushara.com/DarkHeart/RH2/RH.htm)

Fascinating stuff, but the results look a bit... grotesque. Mandelbrot sets of geometrically complicated functions usually can't match elegance and beauty of the z^2 mandelbrot set.

Title: Re: Riemann zeta function as fractal how ?
Post by: flexiverse on March 05, 2015, 07:37:42 PM

Still rather baffled how these are created , is there some code that can be pasted into a modern fractal program to explore this fractal ?

That aside is the most pretty fractal regarded as the Mandelbrot ?

Quote from: laser blaster on March 05, 2015, 06:28:06 PM

Title: Re: Riemann zeta function as fractal how ?
Post by: DarkBeam on March 05, 2015, 07:47:29 PM

You didn't read even the pages? There's an open source software ... http://www.dhushara.com/DarkHeart/Viewers/source/RZ.zip :dink:

Title: Re: Riemann zeta function as fractal how ?
Post by: flexiverse on March 06, 2015, 12:21:41 AM

Of course yes... :embarrass: but I was thinking more of a script/formula
For an existing fractal program like ultra fractal :http://formulas.ultrafractal.com/cgi/formuladb?browse

Not being a mathematical expert or anything, I'm getting the impression Riemann fractal
Is more complicated than the usual fractal.
I mean it doesn't seem to be iterating complex numbers for example....
So I'm guessing this can only be done with a custom program only ???

Quote from: DarkBeam on March 05, 2015, 07:47:29 PM

You didn't read even the pages? There's an open source software ... http://www.dhushara.com/DarkHeart/Viewers/source/RZ.zip :dink:

Title: Re: Riemann zeta function as fractal how ?
Post by: cKleinhuis on March 06, 2015, 12:24:39 AM

there exists a rieman fractal in the ultrafractal database

try this (after downloading the formula collection in uf)

copy and paste the code to ultrafractal render window

Code:

Fractal2 {
fractal:
  title="Fractal2" width=640 height=480 layers=1
  credits="ckleinhuix;3/6/2015"
layer:
  caption="Background" opacity=100 method=multipass
mapping:
  center=0.121659735/-0.2649726022 magn=1.768267
formula:
  maxiter=1000 percheck=off filename="akl.ufm" entry="AL17"
  p_intVal=1/0 p_exponent=2 p__bailout=666 p_pure=yes
inside:
  transfer=none
outside:
  transfer=linear
gradient:
  smooth=yes index=0 color=8716288 index=100 color=16121855 index=200
  color=46591 index=300 color=156
opacity:
  smooth=no index=0 opacity=255
}

Title: Re: Riemann zeta function as fractal how ?
Post by: lkmitch on March 06, 2015, 05:53:41 PM

Mathworld has graphs of the zeta function and an interactive plotter:

http://mathworld.wolfram.com/RiemannZetaFunction.html

The basic function doesn't appear fractal to me.

Title: Re: Riemann zeta function as fractal how ?
Post by: ericr on March 06, 2015, 07:57:47 PM

I hoped that you can make a 3d Zeta function in M3d those will be sublime (but very very complex) : :D

Title: Re: Riemann zeta function as fractal how ?
Post by: DarkBeam on March 06, 2015, 08:58:18 PM

Quote from: ericr on March 06, 2015, 07:57:47 PM

I hoped that you can make a 3d Zeta function in M3d those will be sublime (but very very complex) : :D

Nope... Pure nonsense ;D

Title: Re: Riemann zeta function as fractal how ?
Post by: flexiverse on March 11, 2015, 04:43:32 AM

Quote from: ericr on March 06, 2015, 07:57:47 PM

I hoped that you can make a 3d Zeta function in M3d those will be sublime (but very very complex) : :D

This is exactly what I was thinking!

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Fractal Math, Chaos Theory & Research => (new) Theories & Research => Topic started by: flexiverse on March 05, 2015, 05:13:14 AM