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Are prime numbers regularly ordered?
Volume
145, number
2.3
PHYSICS
ARE PRIME NUMBERS
Z. GAMBA,
REGULARLY
LETTERS
A
2 April I990
ORDERED?
J. HERNANDO
lkpartarnento de Fisica ‘, Comision National de Energia Atbmtca. Av. Ltbertador 8250, 1429 Buenos .4ire.y.Ar,qenttna
and L. ROMANELLI CAERCEM, Julian Alvarez 1218, 1414 Buenos .4ires. Argentina Received 25 September 1989; revised manuscript Communicated by A.R. Bishop
received
4 January
1990; accepted
for publication
5 February
I990
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 chaotic
.4 classical and long standing problem in number theory is the behaviour of the prime number distribution [ 11. Several attempts to find a regular pattern for the prime distribution have been made in the past [2] and, to our knowledge, none of them was successful. 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 contain exactly k primes follow a Poisson distribution ]31. From a physical point of view we thought that if we find that this distribution is chaotic, some nonrigorous answers can be provided. There is also a connection between the prime number distribution and quantum chaos given by the distribution of the imaginary part of the zeros of Riemann’s zeta function i(z), which can be written as a product over all the prime numbers:
Uz)=r-$-$. Berry has extensively ’ Bitnet: SFS7@ARGCNEA2. 106
(1) studied
this problem
while
searching for a model for quantum chaos [ 41. The relationship stems from the Hilbert-Polya conjecture which, as a way to prove Riemann’s hypothesis, suggested the search for a Hermitian operator whose eigenvalues are the imaginary part of the zeros of c(z). Also, the distribution of eigenvalues of GUE (Gaussian unitary ensemble) matrices, prototypical of quantum chaotic systems without temporal invariance [ 41, seems to be the same as that of the zeros of [( I) [ 51. Therefore, whether the distribution of zeros of c(z) does or does not follow the distribution of GUE eigenvalues depends on the distribution of primes. Furthermore. as there is an increasing interest in applying number theory to chaotic dynamics [ 61, we think that it is worthwhile to look into the older number theory problem with the too!s provided by classical chaotic dynamics. The evolution of the power spectrum and Liapunov exponents i, are 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 Ai is positive, we know that the underlying dynamics is chaotic [ 7 1. The series to be analyzed cannot be just the interval between successive prime numbers because this
2 April 1990
PHYSICS LETTERS A
Volume 145, number 2,3
D(x) t
I
I
I
I
moo
4oa
8a.a
boo0
N
Fig. 1. Evolution of the D(X) function; N is the natural number succession.
Fig. 2. Power spectra of the D(x) function
succession is divergent. So, we studied the succession coming from the difference between the primecounting function z(x) and its analytic approximation given by Riemann R(x) (ref. [ I], p. 45 ) . The prime counting function z(x) gives the number of prime numbers less than or equal to x, and R(x) is given by R(x)=Li(x)-f where
Li(&)-f
Li($)-...
,
(2)
Li(x)=
x dx’ an s 2
is the integral logarithm function. We scanned the first 8 x 10’ natural numbers finding 63950 prime numbers. They were used to generate a sequence of 8 x 1O4 equally spaced points of the function D(x)=n(x)-R(x) (equivalent to a stroboscopic sampling). We assume that D(x) pos107
Volume
145, number
2.3
PHYSICS
LETTERS
A
2 April 1990
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.
old
0
I
I
too00
moo
.
I
3oalo
Fig. 3. Liapunov exponent (1,) as a function h! is the sampling point succession.
1
Kale
I
slalo
N
of the sample size;
sesses a stationary distribution, as numerical work for X<~X lOI suggests [8]. Fig. 1 shows the evolution of D(x-) and we analyze its seemingly erratic behaviour by calculating its power spectrum and Liapunov exponent. In fig. 2 the corresponding power spectrum is displayed. The broad band at low frequencies should be noticed, which is a quite strong (necessary but not sufficient ) indication of chaotic behaviour. Fig. 3 depicts the variation of the largest Liapunov exponent with the size of the analyzed succession. 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 20000 points. This Liapunov exponent was calculated by using the method of Eckmann et al. [ 91. This method gives also a minimal embedding dimension of 4 for the unknown subjacent classical dynamic system. The other Liapunov exponents take the following values: 0.00, - 0.04 and - 0.14. The sum of these exponents is -0.07. but we think that our statistics is not sufficient to conclude reliably whether the system is conservative or dissipative.
108
We are greatly indebted to Leticia Hernando ( 12 years old) who, while making her homework, called our attention to this problem. We are also very grateful to Professor J.-P. Eckmann for sending us his Liapunov exponent program [ 91 and to the referee, who was very interested in the work and pointed out to us the relationship with quantum chaos. This work was partially supported by a grant PID 00571/88 by CONICET.
References [ 1] M.R. Schoeder, Number theory in science and communications, Springer series in information sciences (Springer, Berlin, 1986). [2] M. Gardner, Sci. Am. 210(3) (1964) 120. [ 31 P.X. Gallagher, Mathematika 23 (1976) 4. [4] M.V. Berry, Proc. R. Sot. A 413 ( 1987) 183; in Springer lecture notes in physics, Vol. 263. Quantum chaos and statistical nuclear physics, eds. T.H. Seligman and H. Nishioka (Springer, Berlin, 1986) p. 1. [ 51A.M. Odlyzko, Math. Comput. 48 ( 1987) 273. [6] I. Percival and F. Vivaldi, Physica D 25 (1987) 105; D.L. Gonzalez and 0. Piro, Phys. Rev. Lett. 50 ( 1983) 870. [ 71 J.-P. Eckmann and D. Ruelle. Rev. Mod. Phys. 57 (1985) 617: H.G. Schuster, Deterministic chaos ( Physik-Verlag. Weinheim, 1984). [ 81 J.C. Lagarias, VS. Miller and A.M. Odlyzko, Math. Comput. 44 (1985) 537. [9] J.-P. Eckmann, S. Oliffson Kamphorst, D. Ruelle and S. Ciliberto, Phys. Rev. A 34 ( 1986) 497 1.
145, number
2.3
PHYSICS
ARE PRIME NUMBERS
Z. GAMBA,
REGULARLY
LETTERS
A
2 April I990
ORDERED?
J. HERNANDO
lkpartarnento de Fisica ‘, Comision National de Energia Atbmtca. Av. Ltbertador 8250, 1429 Buenos .4ire.y.Ar,qenttna
and L. ROMANELLI CAERCEM, Julian Alvarez 1218, 1414 Buenos .4ires. Argentina Received 25 September 1989; revised manuscript Communicated by A.R. Bishop
received
4 January
1990; accepted
for publication
5 February
I990
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 chaotic
.4 classical and long standing problem in number theory is the behaviour of the prime number distribution [ 11. Several attempts to find a regular pattern for the prime distribution have been made in the past [2] and, to our knowledge, none of them was successful. 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 contain exactly k primes follow a Poisson distribution ]31. From a physical point of view we thought that if we find that this distribution is chaotic, some nonrigorous answers can be provided. There is also a connection between the prime number distribution and quantum chaos given by the distribution of the imaginary part of the zeros of Riemann’s zeta function i(z), which can be written as a product over all the prime numbers:
Uz)=r-$-$. Berry has extensively ’ Bitnet: SFS7@ARGCNEA2. 106
(1) studied
this problem
while
searching for a model for quantum chaos [ 41. The relationship stems from the Hilbert-Polya conjecture which, as a way to prove Riemann’s hypothesis, suggested the search for a Hermitian operator whose eigenvalues are the imaginary part of the zeros of c(z). Also, the distribution of eigenvalues of GUE (Gaussian unitary ensemble) matrices, prototypical of quantum chaotic systems without temporal invariance [ 41, seems to be the same as that of the zeros of [( I) [ 51. Therefore, whether the distribution of zeros of c(z) does or does not follow the distribution of GUE eigenvalues depends on the distribution of primes. Furthermore. as there is an increasing interest in applying number theory to chaotic dynamics [ 61, we think that it is worthwhile to look into the older number theory problem with the too!s provided by classical chaotic dynamics. The evolution of the power spectrum and Liapunov exponents i, are the most elementary tests to be applied to a series of numbers generated by an unknown dynamics in order to search for some hidden regularity. If at least one of the Ai is positive, we know that the underlying dynamics is chaotic [ 7 1. The series to be analyzed cannot be just the interval between successive prime numbers because this
2 April 1990
PHYSICS LETTERS A
Volume 145, number 2,3
D(x) t
I
I
I
I
moo
4oa
8a.a
boo0
N
Fig. 1. Evolution of the D(X) function; N is the natural number succession.
Fig. 2. Power spectra of the D(x) function
succession is divergent. So, we studied the succession coming from the difference between the primecounting function z(x) and its analytic approximation given by Riemann R(x) (ref. [ I], p. 45 ) . The prime counting function z(x) gives the number of prime numbers less than or equal to x, and R(x) is given by R(x)=Li(x)-f where
Li(&)-f
Li($)-...
,
(2)
Li(x)=
x dx’ an s 2
is the integral logarithm function. We scanned the first 8 x 10’ natural numbers finding 63950 prime numbers. They were used to generate a sequence of 8 x 1O4 equally spaced points of the function D(x)=n(x)-R(x) (equivalent to a stroboscopic sampling). We assume that D(x) pos107
Volume
145, number
2.3
PHYSICS
LETTERS
A
2 April 1990
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.
old
0
I
I
too00
moo
.
I
3oalo
Fig. 3. Liapunov exponent (1,) as a function h! is the sampling point succession.
1
Kale
I
slalo
N
of the sample size;
sesses a stationary distribution, as numerical work for X<~X lOI suggests [8]. Fig. 1 shows the evolution of D(x-) and we analyze its seemingly erratic behaviour by calculating its power spectrum and Liapunov exponent. In fig. 2 the corresponding power spectrum is displayed. The broad band at low frequencies should be noticed, which is a quite strong (necessary but not sufficient ) indication of chaotic behaviour. Fig. 3 depicts the variation of the largest Liapunov exponent with the size of the analyzed succession. 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 20000 points. This Liapunov exponent was calculated by using the method of Eckmann et al. [ 91. This method gives also a minimal embedding dimension of 4 for the unknown subjacent classical dynamic system. The other Liapunov exponents take the following values: 0.00, - 0.04 and - 0.14. The sum of these exponents is -0.07. but we think that our statistics is not sufficient to conclude reliably whether the system is conservative or dissipative.
108
We are greatly indebted to Leticia Hernando ( 12 years old) who, while making her homework, called our attention to this problem. We are also very grateful to Professor J.-P. Eckmann for sending us his Liapunov exponent program [ 91 and to the referee, who was very interested in the work and pointed out to us the relationship with quantum chaos. This work was partially supported by a grant PID 00571/88 by CONICET.
References [ 1] M.R. Schoeder, Number theory in science and communications, Springer series in information sciences (Springer, Berlin, 1986). [2] M. Gardner, Sci. Am. 210(3) (1964) 120. [ 31 P.X. Gallagher, Mathematika 23 (1976) 4. [4] M.V. Berry, Proc. R. Sot. A 413 ( 1987) 183; in Springer lecture notes in physics, Vol. 263. Quantum chaos and statistical nuclear physics, eds. T.H. Seligman and H. Nishioka (Springer, Berlin, 1986) p. 1. [ 51A.M. Odlyzko, Math. Comput. 48 ( 1987) 273. [6] I. Percival and F. Vivaldi, Physica D 25 (1987) 105; D.L. Gonzalez and 0. Piro, Phys. Rev. Lett. 50 ( 1983) 870. [ 71 J.-P. Eckmann and D. Ruelle. Rev. Mod. Phys. 57 (1985) 617: H.G. Schuster, Deterministic chaos ( Physik-Verlag. Weinheim, 1984). [ 81 J.C. Lagarias, VS. Miller and A.M. Odlyzko, Math. Comput. 44 (1985) 537. [9] J.-P. Eckmann, S. Oliffson Kamphorst, D. Ruelle and S. Ciliberto, Phys. Rev. A 34 ( 1986) 497 1.