Can Zeta-Function Determine All Fractals or Are Beach Photos Better?

[Sariaht]

Does the Zeta-function provide every fractal their is or should i take beach photoes?

 

[humanino]

Every fractal there is

I am not an expert, but I think it does not make much sens. Fractals can have any real dimension (such as 5.78 for instance). The $$\zeta$$ is defined on the complex plane. How could you produce any fractal with it ?

 

[Sariaht]

I just woundered if my fractal forger can produce any fractal with the weird function Z<-Z^2 + C.

I naturally thought, oh never mind.

 

[Muzza]

Talking about "the" zeta function usually implies the Riemann zeta function, which of course has very little (nothing) to do with fractals. This explains humanino's confusion.

 

[Sariaht]

though i read: "Voronin's theorem on the `Universality'' of Riemann zeta function is shown to imply that Riemann zeta function is a fractal"

I must admit: I made a misstake

 

[Muzza]

It appears as if you are right. Although that stuff is way over my head..

 

[graphic7]

Complex numbers are actually *very* important in fractals, take the Mandelbrot set for instance (along with each Julia Set).

I'm not aware of any fractal that uses the Riemann Zeta function. I wouldn't rule it out as being impossible or impractical. As you probably know, the Riemann Zeta function's most elegant properties are exhibited on the complex plane, and I wouldn't be suprised if a fractal did use the Zeta function.

 

[shmoe]

Voronin's universality is pretty impressive. It does imply that the curve $$\zeta(3/4+it)$$ where $$t\in{\mathbb R}$$ is dense in the complex plane (actually it implies something more general, but that's good enough for now). So it's kind of a space filling curve, but it does cross itself many times. Maybe that's what you are thinking about?

edit-ahh here's a neat paper I'd never seen before, and probably what started this:
http://xxx.lanl.gov/abs/chao-dyn/9406003

 

[Sariaht]

no picture!

I wounder what the fractal looks like. Do you know? They must have made a picture?

 

[graphic7]

I've seen some plots of the Zeta function, probably using the series $$\sum_{x=0}^{\infty}\frac{1}{x^2}$$ at $$\zeta(\frac{1}{2} + it)$$. It's pretty impressive. In the plot you can clearly see how the curve crosses itself and the real line many times (this may be what shmoe is referring to).

One thing that puzzles me, however, is the fact that Voronin only uses $$\zeta(\frac{3}{4} + it)$$. I've always read that the most interesting and elegant properties of Riemann's Zeta function take place at $$\zeta(\frac{1}{2} + it)$$

The picture of what I'm referring to is this:

http://mathworld.wolfram.com/rimg2698.gif

 

[shmoe]

The picture graphic7 posted for $$\zeta(1/2+it)$$is essentially what $$\zeta(3/4+it)$$ looks like (graphic-the series you write isn't related to Zeta at 1/2, in the critical strip Zeta isn't given by a Dirichlet series). It swirls, and swirls, with the swirls getting larger, crossing over itself many, many times. The farther out you go for t, the larger your overall "nest" is in diameter, and the denser it is towards it's center. It's shifted slightly to the right though, and you don't have it passing through 0 like you do with 1/2 (at least not up to any graph that's been done). It's maybe worthwhile to point out that the 3/4 isn't necessary, Voronin's universality implies you can take any $$1/2<\sigma<1$$ and $$\zeta(\sigma+it)$$ will be dense. Actually you get the curve $$(\zeta(\sigma+it),\zeta^{(1)}(\sigma+it),\zeta^{(2)}(\sigma+it),\ldots,\zeta^{(n-1)}(\sigma+it))$$, where the exponents mean various derivatives, is dense in $${\mathbb C}^n$$. If you have maple or matlab, or mathematica you should be able to make some plots easily enough.

Zeta is most interesting in all the critical strip. 1/2 is "extra special" because of the functional equation, but the rest is still interesting. I'm not sure exactly what goes into Voronin's universality, I'm sufficiently interested now that I'll probably take at least a quick peek at his paper, or later generalizations of it (I believe sort of Universality will hold for some more general classes of functions related to Zeta).

I read the article I linked to in my last post. The fractal property he's getting at is just a self similarity. The universality essentially says if you have any analytic function on a disc without zeros then there's a close copy of this function in the right half of the critical strip of Zeta. If you apply this to Zeta itself, say to a big circle to the right of the critical strip where Zeta has no zeros, then Zeta's behavior on this circle is copied inside the critical strip somewhere. This is the self similarity that I guess warrants being called a fractal (I've never studied a rigorous form of chaos, dynamical systems, fractals, etc.).

 

[graphic7]

I ran into a problem with the Dirichlet series several months ago. I wrote a program that was attempting to calculate the Zeta zeros (simple for-loop, few if statements, and lots of iterations). I could never get the Dirichlet series to give a true Zeta zero at 1/2. I ended up ripping off some of Andruw Odlyzko's work, and using his FFT (might have been the Riemann-Siegel formula)to calculate Zeta zeros.

 

[shmoe]

You would run into problems with the Dirichlet series since it's divergent in the critical strip! I do believe the most effiecient methods are based on the Riemann-Siegel formula, though Odlyzko has assuredly improved on the basics quite a bit considering the extent of some of his data.


Here's a nice little animation of Zeta on the critical line:

http://www.math.ubc.ca/~pugh/RiemannZetaComplex/

A few other neat animations can be found here:

http://www.math.ubc.ca/~pugh/index.html