Are Prime Numbers Regularly Ordered?
Departamento de Fisica, Comision Nacional de Energia Atomica, Av.
Libertador 8250, 1429 Buenos Aires, Argentina
CAERCEM, Julian Alvarez 1218, 1414 Buenos Aires, Argentina
This article appeared in Physics Letters A 145, no. 2,3
(2 April 1990), 106-108.
Abstract: The form of the prime number distribution function has
withstood the efforts of all the mathematicians that have considered it.
Here we address this problem with the tools of chaotic dynamics and find
that, from a physical point of view, this distribution function is
The article was not part of the authors'
usual current of research. Hernando's 12 year old daughter
Leticia brought the issue to his attention whilst doing her
homework! As they had been working on certain chaos-related
issues, it occurred to them to apply certain tests to the prime
distribution which are normally applied to physical systems
studied by chaos theorists.
The article begins:
"A classical and long standing problem in number theory is the behaviour
of the prime number distribution .
Several attempts to find a regular pattern for the prime distribution
have been made in the past  and, to our knowledge, none of them was
From a strictly mathematical point of view, this problem was extensively
studied and still remains unsolved. However, some statistical results have
been obtained, e.g. the fractions of intervals which contains exactly
k primes follow a Poisson distribution ."
From a physical point of view we thought that if we find that this
distribution is chaotic, some non-rigorous answers can be provided."
They go on to discuss the connection with quantum chaology, and the study
of the Riemann zeta function by
in the search for a model of quantum
chaos , and a possible proof of the Riemann
hypothesis, based on the "spectral conjecture"
of Hilbert and Pólya. The GUE hypothesis
is also mentioned.
"Furthermore, as there is an increasing interest in applying number theory
to chaotic dynamics , we think that it is worthwhile to look into the
older number theory problem with the tools provided by classical chaotic
The evolution of the power spectrum and
lambdaIare the most elementary tests to be applied to
a series of numbers generated by an unknown dynamics in order to search for
some hidden regularity. If at least one of the lambdaI
is positive, we know the underlying dynamics is chaotic
Rather than analysing the sequence of primes itself, the authors consider
the difference between the prime counting function
analytic approximation function R(x) given by Riemann. This difference
function D(x) shows how the actual distribution of primes fluctuates
around its analytic density. In some sense it is a more direct
representation of the "behaviour" of the primes than
the more obvious . They include a graph of D(x)
as their Figure 1.
Fig.1. Evolution of the D(x) function; N is the
natural number succession
Figure 1 shows the evolution of D(x) and we analyze its seemingly
erratic behaviour by calculating its power spectrum and Liapunov exponent."
Figure 2 shows the corresponding power spectrum. They observe a broad band
at low frequencies, and point out that it is quite a strong (necessary but
not sufficient) indication of chaotic behaviour.
[Figure 3 to be inserted here]
Figure 3 shows the variation of the largest Liapunov exponent with the
size of the analyzed succession. The authors observe that it is
unequivocally positive in all the range and, after an initial increase, a
wide plateau is reached with a convergent value of 0.11 after nearly
20,000 points. They calculated this with the method of Eckmann, et. al.
. It turns out that this method also gives a minimal embedding dimension
of for the unknown underlying classical dynamic system.
The other Liapunov exponents are 0.00, -0.04, and -0.14 to give a sum of
-0.07. The authors safely conclude that their statistics are not sufficient
to conclude reliably whether the system in question is conservative or
"Therefore we can safely conclude that a regular pattern describing the
prime number distribution cannot be found. Also, from a physical point of
view, we can say that any physical system whose dynamics is unknown but
isomorphic to the prime number distribution has a chaotic behaviour."
 M.R. Schroeder, Number theory in science and communications,
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(Springer, Berlin, 1986) p.1.
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