Riemann’s zeta function and the prime series
display a biotic pattern of diversification, novelty, and complexity
Louis H. Kauffman and Hector Sabelli
University of Illinois at Chicago and Chicago Center for Creative
Development, Chicago, IL
kauffman@uic.edu
Here we investigate the processes of
accumulation and rotation in the generation of novelty by examining series
generated by the accumulation of primes and approximations to the primes via
progressive divisibility selections, and series generated by Riemann’s
equation. Riemann’s zeta function is a reencoding of the structure of the
prime numbers. These studies indicate that the
generation of primes is a fundamental example of causal creativity generated by
biotic processes.
1 Introduction
The generation of primes epitomizes the causal
creation of novelty. The prime series, in fact, shows a biotic pattern
[Sabelli, 2007]. Bios is a causallygenerated pattern characterized by
diversification, novelty and complexity [Kauffman and Sabelli, 1998; Kauffman, and Sabelli, 2003; Sabelli, 2005]. The
main tool for measuring bios is the plotting and quantification of isometries
compared to those for shuffled versions of the series [Sabelli and Kauffman, 1999; Sabelli et al, 2005].
Not only the prime number series displays a biotic
pattern, but also the process that
generates the prime series appears to be biotic insofar as it involves
accumulation and periodicity or rotation, which are the two components of
recursions that generate biotic patterns, such as the process equation
[Kauffmann and Sabelli, 1998] and the Schrödinger equation [Sabelli and
Kovacevic, 2006]. Accumulation and rotation correspond to fundamental and
universal components of natural processes; thus radiation involves the
conservation and propagation of energy and the helical pattern of electrical
and magnetic polarity.
The number series also involve accumulation and
rotation like features. Numbers are generated by the fundamental process of
accumulation, the addition of one. Given an integer N, we always have a new
integer N+1 which is different from all the previous integers. This process
alone is deterministic and generates novelty, namely the primes. Furthermore
every integer has a unique factorization as a product of primes, so the novelty
generated by this process is reflected by every integer generated by the
addition process. Each prime is the originator of the infinite set of its
multiples. The proof that there are infinite primes resides in this cumulative
process, as the addition of one to the product of any number of primes is
either a new prime or it has a new prime factor. There are infinitely many
primes. The infinite number of novel primes generates an infinite number of
harmonies (i.e. multiples as in musical otaves).
Numbers are thus generated by two fundamental process
of accumulation, the addition of one and the multiplication by primes. How
these two processes relate is a central issue of number theory and more
generally of natural processes. Multiplication relates directly to rotational
processes. For instance, the numbers on a clock correspond to the multiples of
twelve. Gauss [Gauss (1801)] introduced the generalization of clocks to number
theory in the form of modular number systems.
Here we investigate the
processes of accumulation and rotation in the generation of novelty by
examining series generated by the accumulation of primes and approximations to
the primes generated by progressive divisibility selections, and series
generated by the Riemann’s equation. Riemann’s zeta function is a reencoding
of the structure of the prime numbers.
2 Methods
We study these series with the same methods for analysis used in a
companion article [Sabelli, 2007] to study the primes. We look for complexes
(clusters of isometries separated by interruptions of recurrence) in recurrence
plots, novelty (less recurrence of isometric vectors in the series than in
copies randomized by shuffling) in embedding plots, and the formation of
concentric circles (“Mandala”) in complement plots. Trended series are
detrended by subtracting the local average (ten consecutive members of the
series centered in each point of the series).
3 Primes and Pseudoprimes
Let
P(n) denote the nth prime number. As one detects and isolates the prime numbers
from the series of integers by successively selecting and deleting the
multiples of 2, 3, …P(n), we generate approximations to the primes by
progressive divisibility selections, i.e. all integers except those divisible
by the prime number P(n), which we shall denote as divisibility factor, and all
primes less that P(n) (e.g. divisibility 7 means the series of integers from 2
to N [for some chosen N] when the multiples of 2, 3, 5, and 7 are filtered
out).
Detrended series of primes generate biotic patterns
characterized by complexes in recurrence plots, novelty in embedding plots and
a pattern of concentric circles in complement plots (Figure 1 top). Pseudoprimes
generate periodic patterns characterized by complexes in recurrence plots and
periodic peaks of recurrences in embedding plots for small P(n). For larger
P(n), the series show biotic patterns, which consistently show less recurrence
than their shuffled copies (novelty). As P(n) increases, complexes become
smaller, novelty increases, and there is a larger number of concentric circles
in complement plots. This is because larger differences between consecutive numbers
generate smaller concentric circles. Such pattern of multiple circles indicates, in our view, the
existence of a finite number of rotational generators.
What is not
obvious is why there is a finite number of circles even when plotting primes.
4 Cumulative
series of primes and pseudoprimes
The series A(n) = sin(P(n)), where P(n) is the n^{th
}prime, is chaotic. In contrast, biotic patterns are generated by A(n+1) =
A(n) + sin(P(n)) (figure 2). Note the periodicity (indicated by peaks in the
embedding plot) when the divisibility factor is small. When the divisibility
factor is 7, some periodicity still remains but there also is novelty (number
of isometries less than in shuffled version) and at still larger divisibility
factors the periodicity disappears and the novelty becomes more pronounced.



Figure 1. Recurrence plots (50 embeddings, cutoff
radius 1, 300,000 comparisons) [left], embedding plots [center] where the think
line represents the series and the thin line the shuffled copy, and complement
plots [right] of detrended series of primes and pseudoprimes.
Similar series
constructed by accumulating prime differences after detrending also show
structured recurrence plots and novelty (figure 3). The demonstration of
pattern in the series of differences shows that the series itself is not
stochastic, i.e. it is not generated by patterned changes not random ones.
Figure 2. Recurrence and
embedding plots of the series of sin(prime or pseudoprime) [left two columns]
and of their sum [right two columns].
Figure 3 Recurrence and
embedding plot for the accumulation of prime differences.
Again we see biotic
patterns in approximations to the primes by sieving out only a finite number of
prime multiples from the natural numbers and one can see these approximation
approach the properties of the actual series for the primes as we increase the
divisibility factor. Series generated by A(n+1) = A(n) + sin(P(n)), where P(n)
are pseudoprimes, are periodic when the divisibility factor is small, and
biotic for series generated with a large divisibility factor (figure 2). For
relatively small divisibility factors, the series A(n+1) = sin(P(n)) and A(n+1)
= A(n) + sin(P(n)), are initially more complex (figure 4).
Figure 4. Time series of
A(n+1) = sin(P(n)) [left] and A(n+1) = A(n) + sin(P(n)) [right].
5 Riemann’s
zeta function
The Riemann zeta function is
predated by the following series considered by Euler. The Euler series is the
sum of the reciprocals of the positive integers:
S(N)=1+ 1/2 +
1/3 + 1/4 + …+ 1/N
where N is a positive integer. Euler considers the
limit of this series as N goes to infinity. It turns out that the limit goes to
infinity. The finite series S(N) diverges very slowly as N grows. Nevertheless
it is of interest to write the divergent infinite series E=1+ 1/2 + 1/3 + 1/4 + …, for this formal series is intimately connected with the prime
numbers. Note that for large N, sin(1/N) is approximately 1/N. Thus we can
regard the Euler series as essentially the same as the result of accumulating
sin(1/N) as N ranges over the positive integers. While the series of A(n+1) = sin(P(n) and A(n+1) = A(n) +
sin(P(n) are very complex, the corresponding series for sin(1/P(n)) are not
complex and just become progressively larger. On the other hand if we
accumulate the sine of 1/N^{s}, then complexity^{ }will return
in the form of the behavior of the zeta function that we will discuss below.
Euler
showed the following formula
E = [1/(11/2)] * [1/(11/3)] * [1/(11/5)] * [1/(11/7)]
* ... [1/(11/P(n)]…
where P(n) is a prime. Since
E diverges, there must be an infinite number of primes for this formula to
hold. In this way Euler proved in a new way that there are infinitely many
primes.
Riemann and Euler generalized
the series summing the reciprocals of the natural numbers to a function Zeta(s)
by taking the sum of the reciprocals of the natural numbers raised to the power
of s. That is,
Zeta(s) = 1 + 1/2^{s} + 1/3^{s} + 1/4^{s}
+ ...
The relationship with the
prime numbers still holds, and one has
Zeta(s) = [1/(11/2^{s})] * [1/(11/3^{s})]
* [1/(11/5^{s})] * [1/(11/7^{s})] * ... [1/(11/P(n)^{s}]…
For many values of s, the
Zeta function converges to a specific value. Euler proved that
Zeta(2) = (Pi^{2})/6,
a remarkable result. Riemann
extended the Zeta function to complex values of s. That is, following Riemann,
we can take s = a + bi where i^{2} = 1 and a and b are real numbers. Riemann
then investigated the zeroes of Zeta(s) and found a series of zeroes on the
negative real axis (these are usually called the “trivial” zeros of Zeta), and
he made the conjecture that all the remaining zeroes are of the form
(1/2) + it
for
certain real numbers t. This conjecture remains unproved to this day. It is
called the Riemann Hypothesis. Here we examine series are generated by taking
real and imaginary parts of the zeta function evaluated at the half line s= ½ +
it.
We find biotic behavior
in series generated from the Riemann’s zeta function. Both the real and the
imaginary parts show structure in recurrence plots and novelty in embedding
plots (figure 5).
Figure 5. Recurrence and
embedding plots of the Riemann zeta function at the half line.
One can also
demonstrate evidence of periodicity in complement plots of the Riemann zeta
function, but as the numbers are small are not integers, one must multiply them
by a factor and round them (figure 6).
Figure 6. Complement plots of the real component of
Riemann’s function and of the series of differences between consecutive terms
and their rounded copies after expanding the series by multiplication by 20. N = 2000.
6
Discussion
These studies show how the biotic patterns
observed in the primes are revealed by successive sieving of the integer
series, and that they are also present in Riemann’s zeta function. Further they
show evidence for the role of cumulative and rotational processes. These
conclusions are consistent with ongoing studies on the prime number series and
the Riemann zeta function. Both
periodicity and fractality can be demonstrated in series related to the primes.
Both appear in the statistics of the Goldbach partition [Wang et al., 2006].
Several studies have demonstrated the relevance of periodic functions. There
are periodic oscillations in the histogram of differences of consecutive
primes. The zeros of the Riemann zetafunction can be regarded as harmonic frequencies (via Fourier
transform) in the distribution of primes.
Just as bios is fractal, so is the
zeta function [Woon, 1994]. The spacings
between successive zeroes seem to have a Hurst exponent of about 0.095,
implying antipersistence and high fractal dimension [Shanker, 2006]. While
this is observed in fractional Brownian motion, these features are also present
in bios. Obviously primes are generated causally rather than by the addition of
random changes. The series of differences between consecutive terms of noise is
random. The differences between consecutive terms of a biotic series are
chaotic [Sabelli, 2005]. Chaos phenomena have been found in the difference of
prime numbers [Ares and Castro, 2006]. These authors proposed a theory to
account for the periodic behavior seen in the consecutive differences between
prime numbers that links with the HardyLittlewood conjecture concerning the number of primes in intervals, the statistical mechanics of spin systems, and the
Sierpinski fractal.
Schlesinger [1986] has recast Riemann's Hypothesis
into a probabilistic framework connected to the fractal behavior of a lattice
random walk. We propose instead to cast it as a biotic process generated by
cumulative and rotational processes. The theoretical importance of this
formulation lies to its connection to undulatory physical processes. The
practical importance of this rotational process may be related to Shor’s quantum
computational algorithm for factorization [Shor, 1997].
Assuming that numbers abstract basic
properties of physical action, it is tempting to relate the cumulative and
rotational aspects of these numerical processes to the corresponding properties
of time described by [Gould (1987)]. The
evolution of
spacetime appears to be chaotic [Cornish and Levin, 1997], not random.
Aref'eva and Volovich [2007] have proposed a quantization of the Riemann zeta
function and relate it to classical and quantum field theories as well as to
string cosmological models. Connes [1998] suggests that the distribution of the
Rieman zeroes is in the pattern of a radiative absorption spectrum. Berry and Keating [1999] suggest that the
zeroes of
the zeta function are in correspondence with the energy levels of a quantum
mechanical system.
In summary, the study of primes and related series
indicates that these mathematical processes are fundamental examples of causal
creativity apparently generated by biotic processes of accumulation and
rotation, and that these processes are intimately related to the properties of
physical systems.
Acknowledgements: This research was supported by generous gifts from Ms. Maria
McCormick to the Society for the Advancement of Clinical Philosophy.
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