Zeta functions: main bulb repetition in iteration fractals
In previous posts on this page, and in a recent talk on this subject, I have remarked that the iteration fractal in the neighbourhood of each non-trivial zero of the Riemann zeta function is a scaled and distorted reproduction of the main bulb.
One way of evidencing this phenomenon is to generate the fractal for each starting value, s, in a random section of the complex plane and, for each value of s where the real and imaginary parts of ζ(s) are both within some fixed distance from the origin, to place a pixel at that point with the same colour as the colour ascribed to s in the fractal image.
For example, the following iteration fractal image covers the section of the complex plane between -2 ≤ Re(s) ≤ 13 and 100 ≤ Im(s) ≤ 150, with a resolution of 100 pixels per unit. The image has been rotated 90° clockwise such that the imaginary axis runs from left to right:
The image contains 7,506,501 pixels. For many of the individual starting values, s, the absolute values of the real and imaginary parts of ζ(s) are less than 25. If each qualifying pixel is plotted in a simple x,y plot covering the section of the complex plane between -25 ≤ Re(s) ≤ 25 and -25 ≤ Im(s) ≤ 25, using the colour of the pixel ascribed to the starting value, s, in the above image, the result is as follows:
It is clear that the overall image corresponds to the main bulb of the iteration fractal, but the density of coverage away from the origin decreases rapidly.
The same logic can be applied to Hurwitz zeta functions, ζ(s, a), with rational a.
For example, the following iteration fractal image for the Hurwitz zeta function ζ(s, 0.75) covers the section of the complex plane between -2 ≤ Re(s) ≤ 13 and 100 ≤ Im(s) ≤ 150, with a resolution of 100 pixels per unit. The image has been rotated 90° clockwise such that the imaginary axis runs from left to right:
As before, for many of the individual starting values, s, the absolute values of the real and imaginary parts of ζ(s, 0.75) are less than 25. If each qualifying pixel is plotted in a simple x,y plot covering the section of the complex plane between -25 ≤ Re(s) ≤ 25 and -25 ≤ Im(s) ≤ 25, using the colour of the pixel ascribed to the starting value, s, in the above image, the result is as follows:
Again, it is clear that the overall image corresponds to the main bulb of the iteration fractal. The density of coverage is higher than for the Riemann zeta function.
It is also possible to work in the opposite direction.
For example, the following iteration fractal image for the Hurwitz zeta function ζ(s, 0.75) covers the section of the complex plane between -25 ≤ Re(s) ≤ 25 and 25 ≤ Im(s) ≤ 25, with a resolution of 40 pixels per unit. The image has been rotated 90° clockwise such that the imaginary axis runs from left to right:
The absolute value of the real and imaginary parts of ζ(s, 0.75) for each starting value, s, in a given section of the complex plane may be less 25. If so, we can colour the pixel corresponding to s using the corresponding pixel colour from the above image.
The result for the section of the complex plane covering -2 ≤ Re(s) ≤ 13 and 100 ≤ Im(s) ≤ 150, with a resolution of 100 pixels per unit, is as follows:
The image constitutes a reasonable approximation of the directly calculated fractal (see above) and takes considerably less time to generate. The black areas above and below the image correspond to places where the real and/or imaginary parts of ζ(s, 0.75) are greater than 25.
Another way to visualise this scaling and distortion phenomenon caused by zeta functions is to repeat the previous experiment with an arbitrary geometric design.
For example, the following image corresponds to the section of the complex plane covering -25 ≤ Re(s) ≤ 25 and 25 ≤ Im(s) ≤ 25, with a resolution of 40 pixels per unit. The image has been rotated 90° clockwise such that the imaginary axis runs from left to right:
The following image is formed by calculating ζ(s) for each staring value, s, in the section of the complex plane covering -2 ≤ Re(s) ≤ 13 and 200 ≤ Im(s) ≤ 220 and colouring those pixels where the absolute value of the real and imaginary parts of ζ(s) are less than 25 using the corresponding pixel colour from the above image. The result is as follows:
The overall pattern is preserved with repetition, but it is clear that the green and yellow sections are twisted such that they face upwards rather than their downward orientation in the original image.
Here is the corresponding image for ζ(s, 0.35):
It is clear that, not only are the green and yellow sections twisted above, there is a repetition below where the blue and red sections are twisted such that they face downwards rather than their upward orientation in the original image.
Here is the corresponding image for ζ(s, 0.75):
The repetition in the lower section has persisted. Additional experiments indicate that this repetition below is a feature of all Hurwitz zeta functions ζ(s, a) with rational a.
The same kind of distortion can be applied to any image arbitrarily centred on the origin in the complex plane. The results are particularly disconcerting when the original image is a human face. For example, here is a computer generated image of a human face copied from the website thispersondoesnotexist.com:
Here is the distortion caused by ζ(s, 0.35) in the section of the complex plane covering -2 ≤ Re(s) ≤ 13 and 200 ≤ Im(s) ≤ 220:
Here is the corresponding distortion caused by ζ(s, 0.75):
The inversion / twisting, and upper and lower repetitions that are a feature of Hurwitz zeta functions are easy to discern.









