The Newtonian Orbits of the Riemann Zeta Function:
A Step Towards a Proof of the Riemann Hypothesis

A powerful approach to the study of the zeros of a given function is to investigate the iterative behavior in the complex plane of a companion function whose fixed points coincide with the zeros of the given function. Given a function ƒ(z) there are many possible choices of companion function F(z). Newton's Method, typically introduced in Calculus I, uses the companion function.

Starting with an initial value z_0, we form the sequence of iterations

The resulting sequence of points { z_0, z_1, z_2, z_{3, …} z_n,…} is referred to as the orbit of the initial point z₀. If the companion F (z) is as in Newton's Method then the orbits are called Newtonian.

A typical Newtonian orbit is an infinite sequence of points which, if certain conditions are satisfied, converge to specific point z₀^*. Such an orbit is called convergent. However, a word of caution is in order. For a given function ƒ(z) there may exist many renegade points whose orbits are divergent. If the orbit of z₀ converges to z₀^*, then z₀^* is easily seen to be a fixed point of the companion function F(z). That is F(z₀^*) = z₀^*. This in turn typically implies that z₀^* is a zero of ƒ(z). That is ƒ(z₀^*) = 0.

Newton's Method is a powerful algorithm for locating the zeros of a function of a real variable. However, for this project we decided, by way of experiment, to apply Newton's Method to a complex variable function, that is a function of type ƒ(z) where z = x + iy and where the values of ƒ(z) are themselves complex numbers.

The specific function we chose for the experiment is the famous Riemann zeta function defined, for  (z) > 1, by the formula

z (z) = S 1 / n^z

What makes the Riemann zeta function so famous is the Riemann Hypothesis. To begin with, it is a well established fact that every negative even integer (i.e. …, -6, -4, -2) is a zero of z (z) and that no other real zeros exist. These, the so-called real zeros, are considered only mildly interesting. The Riemann Hypothesis states that every non-real zero of z (z) must lie on the vertical line  (z) = ½. This conjecture, still unproved, is considered by many leading mathematicians to be the most important unsolved problem in modern mathematics.

The function z (z) is known to possess an analytic continuation to the entire complex plane with the exception of a singularity (simple pole) at z = 1. For technical reasons, it is convenient to banish this singularity by defining a closely related auxiliary function

g(z) = (1 - z) z (z)

The function g(z) as defined above possesses essentially the same properties as z (z) but has the advantage of being everywhere defined and differentiable (i.e. holomorphic). In our experiment we applied Newton's method to the companion function

F(z) = z - g(z) / g'(z).

We computed and recorded the orbits of approximately 10,000 initial points selected in the rectangular frame -10 < x < 0 by 5 < y < 55. Limitations on the capacity of MAPLE prevented us from collecting much more data. We plotted the orbits in the rectangular frame -10 < x < 10 by -20 < y < 50. All orbits converging to a specific non-real zero were painted with the same hue. Different non-real zeros were assigned different hues. All orbits converging to real zeros were painted black. All told, the resulting portrait of the Newtonian orbits of the Riemann zeta function consists of approximately 100,000 color-coded points.

Results

We developed two different graphs. One portrayed the real orbits of the function and the other depicted the imaginary orbits. View them here: Real Imaginary

Observations

1. Very few renegade points were encountered. However, the probability of encountering a renegade point seems to increase as  (z) increases. Some or all of these apparent renegade points may in fact be artifacts due to limited computational accuracy.

2. All orbits originating in the band -10 < x < 0 by 0 < y < 8 appear to converge to real zeros only.

3. Every non-real zero appears to be the center of a small finite disk such that every orbit originating within the disk converges to the zero in the center.

4. The orbits appear to align themselves in distinct continuous trails (as evidenced by color). The trails appear not to cross one another.