Mandelbrot Sets Generated from the Riemann Zeta Function

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Mandelbrot Set Generated from the Zeta Function

The Classical Mandelbrot Set

The classical Mandelbrot set is the set of complex numbers c for which the sequence z_n does not diverge to infinity under the recurrence relation

Starting from Costas Simeonidis's GPL-licensed Mandelbrot viewer, I modified the code and added animation. Points inside the set are usually shown in black, but here they are colored according to their complex values.

Full Mandelbrot View
(Iterations: 64)

Drag the mouse to draw a rectangle and zoom in on that region. The default iteration count, or recursion depth, is 16. Click [Start] to animate the process by gradually increasing the iteration count, so you can watch the set take shape step by step.

Zooming in near the origin gives the image below.

Near the Origin
(Iterations: 64)

You can view a spiral structure here.

Mandelbrot Spiral
(Iterations: 256)

The animation smoothly zooms from the full view down into this spiral.

An Exponential-Map Mandelbrot Set

Replacing the quadratic map with an exponential function also produces a Mandelbrot-like set:

The result can be viewed here.

Exponential-Map Overview
(Iterations: 4)

The default iteration count is 4. Press [Start] to animate the set as the iteration count increases.

A magnified view around the origin is shown below.

Exponential Map near the Origin
(Iterations: 64)

You can also view the spiral region here, although rendering may take some time.

Exponential-Map Spiral
(Iterations: 256)

The Riemann Zeta Function

The Riemann zeta function is defined by the series

You can explore its graph here.

Graph of the Zeta Function
(Terms: 16)

Rendering is done with HTML Canvas and Web Workers. The number of terms is used to truncate the series, and the default is 16. Click [Start] to animate the graph as the number of terms increases and to watch the zeros appear.

• The color scale runs from black to white, then blue, cyan, and sky blue.
• The dots along the left edge are the trivial zeros.
• The vertical column of black dots marks the non-trivial zeros.
• The white spot in the center indicates a pole.

A Zeta-Function Mandelbrot Set

Replacing F(z) with the zeta function produces another Mandelbrot-type set:

The full view is available here.

Zeta-Map Overview
(Iterations: 4, Terms: 16)

Use the [Start] buttons for iterations and terms to animate those parameters separately.

This set was introduced in 1998 by Wuon at Cambridge University (arXiv link).

A zoom near the first non-trivial zeros is shown below. Rendering may be slow.

Near the Zeta Zeros
(Iterations: 64, Terms: 8)

You can view the spiral here. This view is especially slow to render.

Zeta-Map Spiral
(Iterations: 256, Terms: 8)

A flower-like structure appears in this region.

Comparing the Mandelbrot Variants

Although the maps are different, these Mandelbrot-type sets display surprisingly similar patterns.

Comparison of Mandelbrot-Type Sets

F(z)	z2	ez	ζ(z)
Full View
Near Origin/ Zeros
Spiral

The resemblance may be related to the following facts:

• The exponential function can be expressed as a power series.

• The zeta function can be rewritten as a sum of exponentials.

Could There Be a Physical Meaning?

Each point in a Mandelbrot set either remains bounded or diverges to infinity. In a loose analogy, the universe may also either expand forever or eventually collapse. Perhaps each point could be imagined as representing a different universe.

How the Zeta Function Is Computed

Because the naïve series converges slowly, the program switches between several formulas as needed:

• Euler-Maclaurin asymptotic expansion
• Knopp-Hasse formula
• Peter Borwein's acceleration method
• The Riemann functional equation, when necessary

Accuracy tests for these formulas are summarized here:

Accuracy Comparison of the Algorithms

User Guide

Basic Controls

• Drag to draw a rectangle and zoom in.
• The Iteration button changes the recursion depth.
• The Terms button changes the number of series terms.
• [Reset] returns to the initial view.

Animation Controls

• [Start] begins an animation that gradually increases the number of iterations or terms.
• [Stop] pauses the animation.

Other Features

• [Zoom out] zooms back out.
• [Sizes] changes the canvas size.
• [PNG] saves the current image as a PNG file.
• [Animation] starts an automatic zoom from the current position.

• Zeta Functions and Bernoulli Numbers: Infinite Sums and Regularization (2015/4/25)
• Do Imaginary Numbers Exist? A Matrix View (2015/7/5)
• Actual Infinity and Potential Infinity: Two Views of Infinity (2015/8/3)

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