The Newtonian Orbits of the Riemann Zeta Function:
A Step Towards a Proof of the Riemann Hypothesis
Nabil Guffey
and Nelson Petulante
Bowie State University Mathematics Department
Bowie, Maryland
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.
F (z) = z  ƒ(z) / ƒ '(z)
Starting with an initial value z_{0, }we form the
sequence of iterations
z_{1} = F (z_{0})
z_{2} = F (z_{1})
z_{3} = F (z_{2})
.
.
.
z_{n} = F (z_{n 
1})
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
z_{0}. 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
z_{0}^{*}. 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 z_{0} converges to
z_{0}^{*}, then z_{0}^{*} is
easily seen to be a fixed point of the companion function F(z).
That is F(z_{0}^{*}) =
z_{0}^{*}. This in turn typically implies that
z_{0}^{*} is a zero of ƒ(z).
That is ƒ(z_{0}^{*}) = 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 Â (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
socalled real zeros, are considered only mildly interesting. The Riemann
Hypothesis states that every nonreal zero of z (z) must lie
on the vertical line Â (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 nonreal zero were painted with the same hue. Different nonreal 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 colorcoded 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
 Very few renegade points were encountered. However, the probability of
encountering a renegade point seems to increase as Â
(z) increases. Some or all of these apparent renegade points may in
fact be artifacts due to limited computational accuracy.
 All orbits originating in the band 10 < x < 0 by 0 <
y < 8 appear to converge to real zeros only.
 Every nonreal 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.
 The orbits appear to align themselves in distinct continuous trails (as
evidenced by color). The trails appear not to cross one another.
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