System synergies: Chaotic or predictable?

S.C. Woon, "Riemann zeta function is a fractal" (preprint 06/94)

"[We] infer three corollaries from Voronin’s theorem [on the ’universality’ of the Riemann zeta function]. The first is interesting, the second is a strange and amusing consequence, and the third is ludicrous and shocking (but a consequence nevertheless)."

"Corollary 1 ("interesting") Riemann zeta function is a fractal."

Woon’s innovation here is to devise analytic function f based on the zeta function itself (involving translations and rescalings), in order to show that zeta replicates its own behaviour infinitely often at all scales.

"Since ao can be arbitrarily chosen, there are self-similarities at all scales. Therefore, Riemann zeta function is a fractal."

He goes on to show that you can choose f to be based on the zeta function not only via translation and dilation, but also invovling rotation and reflection. The result is that we have self-similarities between discs at different scales and orientations."

"Riemann zeta function is fractal in the sense that the Mandelbrot set is fractal (self-similarities between a region bounded by a closed loop C and other regions bounded by closed Cm’ of the same shape at smaller scales and/or orientations). The fractal property of zeta is not "infinitely recursive" as in Koch snowflake. Such infinite recursions in a function will render the function non-differentiable, whereas zeta is infinitely differentiable. So, the manifold of zeta function is not of fractal dimension."

"All Dirichlet L-functions are also fractal. This follows from the remark following Voronin’s theorem in Voronin’s paper."

"Corollary 2 ("strange and amusing") Riemann zeta function is a ’library’ of all possible smooth continuous line drawings in a plane."

Imagine all the ways an analytic function can map a line segment, say the vertical diameter of |z| < 1/4, into the complex plane. Woon points out that every imaginable kind of shape (an outline Mickey Mouse is used as an example) can be drawn in this way. It then follows from the Universality Theorem that the zeta function’s behaviour on segments of Re[s] = 3/4 (or any other line between 1/2 and 1) can replicate any such shape.

"Corollary 3 ("ludicrous and shocking") Riemann zeta function is a concrete "representation" of the giant book of theorems referred to by Paul Halmos."

Woon explains that you can represent arbitrarily long Morse code messages as oscillating curves representing ’signals’. Every possible one of these messages is reproducible to within a workable accuracy by the zeta function. So the entire Encylopedia Britannica could be deduced as a Morse Code transmission encoded as a wave which was the image of a vertical segment of length 1/2 on Re[s] = 3/4 under the Riemann zeta function.

"So... the entire human knowledge are already encoded in zeta function."

"Hence, Riemann zeta function is probably one of the most remarkable functions because it is a concrete "representation" (in group theory sense) of "the God’s giant book of theorems" that Paul Halmost spoke of - all possible theorems and texts are already encoded in some form in Riemann zeta function, and repeated infinitely many times. Although a white noise function and an infinite sequence of random digits are also concrete "representations", Riemann zeta function is not white noise or random but well-defined.

Alternatively, from the point of view of information theory, even though Riemann zeta function is well-defined, its mappings in the right half of the critical strip are random enough to encode arbitrary large amount of information - the "entropy" of its mapping is infinite.

Example This article is also encoded somewhere in Riemann zeta function as it is being written!"

S.C. Woon, "Fractals of the Julia and Mandelbrot sets of the Riemann zeta function" (preprint 12/98)

"Computations of the Julia and Mandelbrot sets of the Riemann zeta function and observations of their properties are made. In the appendix section, a corollary of Voronin’s theorem is derived and a scale-invariant equation for the bounds in Goldbach conjecture is conjectured."