(Retired) Email:
selvam@ip.eth.net
Web site : http://webspace.webring.com/people/oa/amselvam
Abstract
Recent studies indicate a close association between the distribution of prime numbers and quantum mechanical laws governing the subatomic dynamics of quantum systems such as the electron or the photon. Number theoretical concepts are intrinsically related to the quantitative description of dynamical systems of all scales ranging from the microscopic subatomic dynamics to macroscale turbulent fluid flows such as the atmospheric flows. It is now recognised that Cantorian fractal spacetime characterise all dynamical systems in nature. A cell dynamical system model developed by the author shows that the continuum dynamics of turbulent fluid flows consist of a broadband continuum spectrum of eddies which follow quantumlike mechanical laws. The model concepts enable to show that the continuum real number field contains unique structures, namely prime numbers which are analogous to the dominant eddies in the eddy continuum in turbulent fluid flows. In this paper it is shown that the prime number frequency spectrum follows quantumlike mechanical laws.
1. Introduction
The continuum real number field(infinite
number of decimals between any two integers) represented as Cartesian coordinates
[ Mathews, 1961; Stewart and Tall,1990; Devlin,1997; Stewart,1998]
is the basic computational tool in the simulation and prediction of the
continuum dynamics of real world dynamical systems such as fluid flows,stock
market price fluctuations, heart beat patterns, etc. Till the late 1970s
mathematical models were based on Newtonian continuum dynamics with implicit
assumption of linearity in the rate of change with respect to (w.r.t) time
or space of the dynamical variable under consideration. The traditional
mathematical model equations were of the form
Constant value was assumed for the rate of change
of the variable X_{n} at computational step
n
and infinitesimally small time or space intervals dt . Equation(1)
will be linear and can be solved analytically provided the rate of change
is constant. However, dynamical systems in nature exhibit irregular fluctuations
and therefore the assumption of constant rate of change fails and Equation(1)
does not have analytical solution. Numerical solutions are then obtained
for discrete(finite) spacetime intervals such that the continuum dynamics
of Equation(1) is now computed as discrete dynamics given by
Numerical solutions obtained
using Equation(2), which is basically a numerical integration procedure,
involve iterative computations with feedback and amplification of roundoff
error of real number finite precision arithmetic. The Equation(2) also
represents the relationship between continuum number field and embedded
discrete(finite) number fields. Numerical solutions for nonlinear dynamical
systems represented by Equation 2 are sensitively dependent on initial
conditions and give apparently chaotic solutions, identified as deterministic
chaos . Deterministic chaos therefore characterise the evolution
of discrete(finite) structures from the underlying continuum number field.
Historically , sensitive depence on initial conditions of nonlinear dynamical
systems was identified nearly a century ago by Poincare (Poincare,
1892) in his study of three body problem,namely the sun, earth and the
moon . Nonlinear dynamics remained a neglected area of research till the
advent of electronic computers in the late 1950s. Lorenz, in 1963 showed
that numerical solutions of a simple model of atmospheric flows exhibited
sensitive dependence on initial conditions implying loss of predictability
of the future state of the system. The traditional nonlinear dynamical
system defined by Equation 2 is commonly used in all branches of science
and other areas of human interest. Nonlinear dynamics and chaos
soon(by 1980s) became a multidisciplinary intensive field of research (Gleick,
1987). Sensitive dependence on initial conditions imply longrange spatiotemporal
correlations. The irregular fluctuations of real world dynamical syatems
also exhibit such nonlocal connections manifested as fractal or
selfsimilar
geometry to the spatiotemporal evolution. The universal symmetry of selfsimilarity
ubiquitous to dynamical systems in nature is now identified as selforganized
criticality (Bak,Tang and Wiesenfeld, 1988). A symmetry of some figure
or pattern is a transformation that leaves the figure invariant, in the
sense that, taken as a whole it looks the same after the transformation
as it did before, although individual points of the figure may be moved
by the transformation(Devlin,1997 ). Selfsimilar structures have internal
structure which resemble the whole.
The spatiotemporal organization
of a hierarchy of selfsimilar spacetime structures is common to real world
as well as the numerical models(Equation 2) used for simulation . A substratum
of continuum fluctuations selforganizes to generate the observed
unique hierarchical structures both in real world and the continuum number
field used as the tool for simulation. A cell dynamical system model
developed by the author [Mary Selvam, 1990; Selvam and Suvarna Fadnavis,1998,1999a,b]
for turbulent fluid flows shows that selfsimilar (fractal) spacetime fluctuations
exhibited by real world and numerical models of dynamical systems are signatures
of quantumlike mechanics. The model concepts are applicable to the emergence
of unique prime number spectrum from the underlying substratum of
continuum real number field.
Recent studies indicate a close association between number theory in mathematics, in particular, the distribution of prime numbers and the chaotic orbits of excited quantum systems such as the hydrogen atom [Cipra, 1996]. Mathematical studies indicate that cantorian fractal spacetime characterises quantum systems[Nottale, 1989; Ord, 1983; El Naschie, 1993].
2. Model Concepts
A summary of the important results
of the cell dynamical system model results for turbulent fluid flows [Mary
Selvam, 1990; Selvam and Suvarna Fadnavis, 1998,1999a,b] which are applicable
to the present study are given in the following.
Based on Townsend’s [Townsend,
1956] concept that large eddies are envelopes of enclosed turbulent eddy
circulations (Figure 3), the relationship between root mean square (r.m.s.)
circulation speeds W and
w_{*} respectively
of large and turbulent eddies of respective radii R and r
is given as
The dynamical evolution of
spacetime fractal structures can be quantified in terms of ordered energy
flow between fluctuations of all scales described by mathematical functions
which occur in the field of number theory. The quantumlike chaos
in atmospheric flows can be quantified in terms of the following mathematical
functions / concepts: (a) The fractal structure of the continuum flow pattern
is resolved into an overall logarithmic spiral trajectory with the quasiperiodic
Penrose
tiling pattern for the internal structure and is equivalent to a hierarchy
of vortices(Figure 1).
Figure 1 : The quasiperiodic Penrose tiling pattern with fivefold symmetry traced by the small eddy circulations internal to dominant large eddy circulation in turbulent fluid flows.
Historically, the British
mathematician Roger Penrose discovered in 1974 the quasiperiodic
Penrose
tiling pattern, purely as a mathematical concept. The fundamental investigation
of tilings which fill space completely is analogous to investigating the
manner in which matter splits up into atoms and natural numbers split up
into product of primes. The distiction between periodic and aperiodic tilings
is somewhat analogous to the distinction between rational and irrational
real numbers, where the latter have decimal expansions that continue forever
, without settling into repeating blocks [Devlin, 1997]. Even earlier
Kepler
saw a fundamental mathematical connection between symmetric patterns and
'space filling geometric figures' such as his own discovery , the rhombic
dodecahedron , a figure having 12 identical faces [ Devlin,
1997]. The quasiperiodic Penrose tiling pattern has fivefold symmetry
of the dodecahedron. Recent studies[Seife,1998] show that in a strong magnetic
field, electrons swirl around magnetic field lines, creating a vortex.
Under right conditions, a vortex can couple to an electron, acting as a
single unit.
2.1 Model Predictions
(a) Atmospheric flows trace an overall logarithmic spiral trajectory R_{o}R_{1}R_{2}R_{3}R_{4}R_{5} with the quasiperiodic Penrose tiling pattern for the internal structure (Figure 1).
(b) Conventional continuous periodogram power spectral analyses of such spiral trajectories will reveal a continuum of periodicities with progressive increase dq in phase angle q (theta) as shown in Figure 2.
Figure 2: The equiangular logarithmic spiral given by (R/r) = e^{aq} where aandq are each equal to 1/z for each length step growth. The eddy length scale ratio z is equal to R/r . The crossing angle a is equal to the small increment dq in the phase angle q . Traditional power spectrum analysis will resolve such a spiral flow trajectory as a continuum of eddies with progressive increase dq in phase angle q .
(c) The broadband power spectrum will have embedded dominant wavebands(R_{o}OR_{1},R_{1}OR_{2},R_{2}OR_{3},R_{3}OR_{4},R_{4}OR_{5},etc.) the bandwidth increasing with period length(Figure 1). The peak periods E_{n} in the dominant wavebands will be given by the relation
where t
is the golden mean equal to {1+sqrt(5)}/2
[approximately equal to
1.618 ] and T_{s}
, the primary perturbation time period ; for example , T_{s}
is the annual cycle (summer to winter) of solar heating in a study of atmospheric
interannual variability. The peak periods E_{n }
are superimposed on a continuum background . For example, the most striking
feature in climate variability on all time scales is the presence of sharp
peaks superimposed on a continuous background [Ghil,1994].
(d) The ratio r/R also represents the increment dq in phase angle q (Equation 3 and Figure 2) and therefore the phase angle q represents the variance [ Mary Selvam, 1990 ] . Hence, when the logarithmic spiral is resolved as an eddy continuum in conventional spectral analysis, the increment in wavelength is concomitant with increase in phase. The angular turning, in turn, is directly proportional to the variance( Equation 3) Such a result that increments in wavelength and phase angle are related is observed in quantum systems and has been named 'Berry's phase' [Berry ,1988]. The relationship of angular turning of the spiral to intensity of fluctuations is seen in the tight coiling of the hurricane spiral cloud systems.
(e) The overall logarithmic spiral flow structure is given by the relation
where the constant k is the steady
state fractional volume dilution of large eddy by inherent turbulent eddy
fluctuations . The constant k is equal
to 1/t
^{2}
(=0.382) and is identified as the universal constant for
deterministic chaos in fluid flows. Since k is less than
half, the mixing with environmental air does not erase the signature of
the dominant large eddy , but helps to retains its identity as a stable
selfsustaining solitonlike structure The mixing of environmental
air assists in the upward and outward growth of the large eddy. The steady
state emergence of fractal structures is therefore equal to
1/k = 2.62
Logarithmic wind profile relationship
such as Equation 5 is a longestablished(observational) feature of atmospheric
flows in the boundary layer, the constant k, called the Von
Karman ’s constant has the value equal to 0.38 as determined from observations
[ Wallace and Hobbs , 1977] .
In Equation 5, W
represents the standard deviation of eddy fluctuations, since W
is computed as the instantaneous r.m.s.(root mean square) eddy perturbation
amplitude with reference to the earlier step of eddy growth. For two successive
stages of eddy growth starting from primary perturbation w_{*
},
the ratio of the standard deviations W_{n+1}
and W_{n} is given from Equation 5 as (n+1)/n.
Denoting by s
the standard deviation of eddy fluctuations
at the reference level (n=1) the standard deviations of eddy
fluctuations for successive stages of eddy growth are given as integer
multiple of s
, i.e., s
, 2s
, 3s
, etc., and correspond respectively to
statistical normalized standard deviation t = 0,1,2,3, etc.
The conventional power spectrum
plotted as the variance versus the frequency in loglog scale will now
represent the eddy probability density on logarithmic scale versus the
standard deviation of the eddy fluctuations on linear scale since the logarithm
of the eddy wavelength represents the standard deviation , i.e. the r.m.s.
value of eddy fluctuations (Equation 5). The r.m.s. value of eddy fluctuations
can be represented in terms of statistical normal distribution as follows.
A normalized standard deviation t = 0 corresponds to cumulative
percentage probability density equal to 50 for the mean value
of the distribution. Since the logarithm of the wavelength represents the
r.m.s. value of eddy fluctuations the normalized standard deviation t
is defined for the eddy energy as
t = (log L / log T_{50} )  1
where L is the period
in years and T_{50} is the period up
to which the cumulative percentage contribution to total variance is equal
to 50 and t = 0 .
The variable log T_{50}
also represents the mean value for the r.m.s. eddy fluctuations and is
consistent with the concept of the mean level represented by r.m.s. eddy
fluctuations. Spectra of time series of any dynamical system, for example,
meteorological parameters when plotted as cumulative percentage contribution
to total variance versus
t should follow the model
predicted universal spectrum. The literature shows many examples of pressure,wind
and temperature whose shapes display a remarkable degree of universality
[Canavero and Einaudi,1987].
The theoretical basis for
formulation of the universal spectrum is based on the Cental Limit Theorem
in Statistics, namely, if an overall random variable is the sum of
very many elementary random variables, each having its own arbitrary distribution
law, but all of them being small, then the distribution of the overall
random variable is Gaussian [Ruhla and Barton,1992] . Therefore,
when the spectra of spacetime fluctuations of dynamical systems are plotted
in the above fashion, they tend to closely (not exactly) follow cumulative
normal distribution.
The period T_{50}
up to which the cumulative percentage contribution to total variance is
equal to 50 is computed from model concepts as follows.
The power spectrum, when
plotted as normalized standard deviation t versus cumulative
percentage contribution to total variance represents the statistical normal
distribution(Equation 8), i.e. the variance represents the probability
density. The normalized standard deviation value 0 corresponds
to cumulative percentage probability density P equal
to
50 from statistical normal distribution characteristics.
Since t represents the eddy growth step n
(Equation 7), the dominant period T_{50}
upto which the cumulative percentage contribution to total variance is
equal to 50 is obtained from Equation 4 for value
of n equal to 0 . In the present
study of periodicities in prime number spacing intervals, the primary perturbation
time period T_{s} is equal to the unit number
class interval and T_{50} and is obtained as
T_{50} = (2+t )t ^{0 } ~ 3.6 spacing interval between two adjacent primes
Prime numbers
with spacing intervals up to 3.6 or approximately 4
contribute upto 50% to the total variance. This model
prediction is in agreement with computed value of T_{50}(Section
3.3 ).
2.2 Applications of model concepts to prime number distribution
The incorporation of Fibonacci
mathematical series, representative of ramified bifurcations, indicates
ordered growth of fractal patterns. The fractal patterns are shown to result
from the cumulative integration of enclosed small scale fluctuations. By
analogy it follows that the continuum number field when computed
as the integrated mean over successively larger discrete domains, also
follows the quasiperiodic Penrose tiling pattern. It is shown in
the following that the steady state emergence of progressively larger fractal
structures incorporates unique primary perturbation domains of progressively
increasing total number equal to z/lnz where
z,
the length step growth stage is equal to the length scale ratio of large
eddy to turbulent eddy. In number theory , prime numbers are
unique numbers and the prime number theorem(PNT) states that
z/lnz
gives approximately the number of primes less than or equal to z
[Rose, 1995] . The model also predicts that z/lnz represents
the normalized cumulative variance spectrum of the eddies and this spectrum
follows statistical normal distribution. The important result of
the study is that the
prime number spectrum is the same as the eddy
energy spectrum for quantumlike chaos in atmospheric flows and the spectra
follow the universal inverse power law form of the statistical normal distribution.
Historically, the PNT
was postulated just before 1800 by both Legendre (1798)and Gauss
(1791in a personal communication) on numerical evidence and it was finally
established by Hadamard and (independently) de la Vallee Poussin
in 1896. The PNT states that if p(z)
is the number of primes p which satisfy 2<= p <= z
then p(z)
is approximately equal to z/ln z where ln represents the
natural logarithm (Rose, 1995; Allenby and Redfern,1989 )
The cell dynamical model concepts and its application to the evolution of prime number spectrum is explained in the following.
Large eddies are envelopes of
enclosed turbulent eddy circulations, the relationship between root mean
square (r.m.s.) circulation speeds W and
w_{*}
respectively of large and turbulent eddies of respective radii R
and r is given as (Equation 3)
In number field domain, the above equation can be visualized as follows. The r.m.s. circulation speeds W and w_{*} are equivalent to units of computations of respective yardstick lengths R and r. Spatial integration of w_{*} units of a finite yardstick length r, i.e. a computational domain w_{*} r, results in a larger computational domain WR [Mary Selvam, 1993]. The computed domain WR is larger than the primary domain w_{*} r because of uncertainty in the length measurement using a finite yardstick length r, which should be infinitesimally small in an ideal measurement. The continuum number field domain(Catesian coordinates ) may therefore be obtained from successive integration of enclosed finite number field domains as shown in Figure 3. Cartesian coordinates represent the complex number field. Historically, Gauss(1799) clearly regarded a complex number as a pair of real numbers. The idea was originally stated in a little known work of a Danish surveyor Wessel(1797) and later by Gauss. In 1806, the French mathematician Argand described a complex number x+iy as a point in the plane and this description was given the name 'Argand Diagram' [ Stewart and Tall, 1990].
Figure 3: Visualisation of the formain of large eddy (ABCD) as envelope enclosing smaller scale eddies. By analogy, the continuum number field domain(Cartesian coordinates ) may also be obtained from successive integration of enclosed finite number field domains.
The above visualization will help apply concepts developed for continuum atmospheric flow dynamics to evolution of structures in real number field continuum such as the distribution of prime numbers, as explained in the following.
Fractal structures emerge in atmospheric flows because of mixing of environmental air into the large eddy volume by inherent turbulent eddy fluctuations. The steady state emergence of fractal structures A is equal to [Selvam and Suvarna Fadnavis, 1999a,b]
The spatial integration of enclosed turbulent eddy circulations as given in Equation(3) represents an overall logarithmic spiral flow trajectory with the quasiperiodic Penrose tiling pattern(Figure 1) for the internal structure [Selvam and Suvarna Fadnavis, 1999a,b] and is equivalent to a hierarchy of vortices ( Section 2 above). The incorporation of Fibonacci mathematical series, representative of ramified bifurcations indicates ordered growth of fractal patterns and signifies nonlocal connections characteristic of quantumlike chaos. By analogy, the means of ensembles of successively larger number field domains follow a logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern(Figure 1) for the internal structure.
The logarithmic flow structure is given by the relation (Equation 5).
where z is equal to the eddy length
scale ratio
R/r and k is equal to the steady
state fractional volume dilution of large eddy by turbulent eddy fluctuations
and is given as
The steady state emergence of fractal structure
A
is
The outward and upward growing large eddy carries
only a fraction f of the primary perturbation equal to
because the fractional outward mass flux of primary perturbation equal to W/w_{*} occurs in the fractional turbulent eddy cross section r/R.
from equation (5)
from equation (10)
from equation (3)
Therefore
from equation (11)
In atmospheric flows a fraction
equal to f of surface air is transported upward to level
z
and represents the upward transport of moisture which condenses as liquid
water content in clouds, and also aerosols of surface origin. The observed
vertical profile of liquid water content inside clouds is found to follow
the f distribution [Mary Selvam, 1990]. The vertical profile
of aerosol concentration in the atmosphere also follows the f
distribution [Sikka et al , 1988]. The fraction
f
carries the unique signature of surface air (primary perturbation) at the
level z.
The f distribution represents, at level z , the signature of unique primary perturbation originating from the underlying substratum. The f distribution therefore corresponds to the cumulative prime number density distribution corresponding to number z .
Therefore the ratio P equal to A/f gives the number of units of the unique domain of surface air at level z.
In number theory, the
Prime
Number Theorem states that z/ln z where
ln
is the natural logarithm , represents approximately the number of primes
less than or equal to z. Prime numbers are unique
numbers, i.e. which cannot be factorized [Stewart, 1996]. Therefore P
represents the cumulative unique domain lengths of the primary perturbation
carried up to the level z .
In the next Section(3.0) the
following model predictions(Section 2.0) are verified.
(a) The f distribution
represents the actual and computed prime number density distributions.
(b) The power spectra(variance
and phase) of prime number distribution follows the universal and unique
inverse power law form of the statistical normal distribution. Inverse
power law form for power spectra signify selfsimilarity or longrange correlations
inherent to the eddy continuum.
(c) The broadband eddy continuum
exhibits dominant periodicities in close agreement with model predicted
peridicities (Equation 4).
(d) The variance and phase spectra
follow each other closely, particularly for the dominant eddies, thereby
exhibiting 'Berry's phase' characterising quantum systems.
3. Data and Analysis
The actual prime number tables (the first 1000 primes) were obtained from the web site: http://www.utm.edu/research/primes. The first 1000 prime numbers were used for the study. The prime numbers were also computed using the Prime Number Theorem proposed in 1799 by Gauss , namely, the total number of primes p(z) equal to or less than the number z is aproximately equal to z/ln z . The computed prime number density distribution is equal to 1/ln z .
The computed f distribution(Equation 12), the actual prime number density distribution and the computed prime number density distributions are shown in Figure 4.
Figure 4: The cumulative prime number(actual) density and the corresponding f distribution have a maximum approximately equal to 0.6 for the number z equal to 2p which represents one complete eddy cycle . The eddy length scale ratio z represents the phase for the eddy continuum dynamics in turbulent fluid flows. A complete dominant eddy cycle(z = 2p ) is a selfsustaining solitonlike structure.
The
shape of the actual prime number density distribution is close to and resembles
f
distribution. Further , the maximum value ( approximately equal to 0.6)
for these two distributions occurs for z value equal to 2p
. The eddy length scale ratio
z represents the phase (Section 2 ) and therefore the maximum
values for f and also (by analogy), for the prime number
distributions occur for one complete cycle of eddy circulation . Such a
closed selfsustaining circulation is similar to a
soliton
, a stable selfsustaining eddy structure.
3.1 The Frequency Distributions of Prime numbers, f distribution and the statistical normal distribution
The values of actual prime number distribution, the corresponding values computed using the relation z/lnz (Prime Number Theorem) which give the number of primes less than or equal to z and the f distribution follow statistical normal distribution as described in the following. The frequency distributions were computed in terms of the normalised standard deviation as explained in the following for prime number(calculated) distribution . The number of primes P equal to or less than z are calculated for a range of n values from x_{1 }= z_{1} to x_{n }= z_{n} . The cumulative percentage number of primes P_{c} is calculated as equal to (P_{m} / P_{n})*100 where m = 1,2,...n for each class interval X = ( x_{m}+x_{m+1 }) / 2. The number of primes P_{t} = P_{m+1}  P_{m} in each class interval X is also calculated. The normalized standard deviate t is then equal to (Xbar  X) / s where Xbar is the mean of the prime number distribution. The corresponding standard deviation of the X versus P_{t } distribution is then calculated as equal to s .
The prime number(actual and computed) frequency distribution and also the corresponding f distribution for values of z from 3 to 1000 at unit intervals are shown in Figure 5. The statistical normal distribution is also plotted in the Figure 5. It is seen that the prime number(actual and computed) distributions and the corresponding f distribution closely follow statistical normal distribution.
Figure 5: Prime number(actual and computed) distribution and corresponding f distribution follow closely the statistical normal distribution.
3.2 Spectra of prime number distribution
In the quantumlike chaos in atmospheric flows the function z/lnz represents the variance spectrum of the fractal structures as shown below.
The length scale ratio z equal to R/r represents the relative variance (Equation 3). The relative upward mass flux of primary perturbation equal to W/w_{*} is proportional to lnz (Equation 5). Therefore z/lnz represents the cumulative variance normalized to upward flow of primary perturbation. The cumulative variance or energy spectrum of the eddies is therefore represented by z/lnz distribution.
By concept (Equation 3) large
eddies are but the integrated mean of inherent turbulent eddies and therefore
the eddy energy spectrum follows statistical normal distribution according
to the Central Limit Theorem (Section 2). The prime number spectrum
which is equivalent to the variance (energy) spectrum of eddies follows
statistical normal distribution as seen in Figure 5 shown above.
Earlier studies using various meteorological data sets have shown that
atmospheric eddy energy spectrum follow statistical normal distribution
[Selvam and Suvarna Fadnavis, 1998].
3.3 Power Spectral Analysis: Analyses Techniques, Data and Results
The broadband power spectrum of spacetime fluctuations of dynamical systems can be computed accurately by an elementary, but very powerful method of analysis developed by Jenkinson (1977) which provides a quasicontinuous form of the classical periodogram allowing systematic allocation of the total variance and degrees of freedom of the data series to logarithmically spaced elements of the frequency range (0.5, 0). The periodogram is constructed for a fixed set of 10000(m) periodicities L_{m } which increase geometrically as L_{m}=2 exp(Cm) where C=.001 and m=0, 1, 2,....m . The data series Y_{t} for the N data points was used. The periodogram estimates the set of A_{m}cos(2pn_{m}Sf_{m}) where A_{m}, n_{m} and f_{m} denote respectively the amplitude, frequency and phase angle for the m^{th} periodicity and S is the time or space interval . In the present study the frequency of occurrence of primes in unit number class intervals ranging from 3 to 1000 was used. The cumulative percentage contribution to total variance was computed starting from the high frequency side of the spectrum. The period T_{50} at which 50% contribution to total variance occurs is taken as reference and the normalized standard deviation t_{m} values are computed as (Equation 8).
t_{m} = (log L_{m} / log T_{50})1
The cumulative percentage contribution to total variance, the cumulative percentage normalized phase (normalized with respect to the total phase rotation) and the corresponding t values were computed . The power spectra were plotted as cumulative percentage contribution to total variance versus the normalized standard deviation t as given above . The period L is in units of number class interval which is equal to one in the present study. Periodicities up to T_{50} contribute up to 50% of total variance. The phase spectra were plotted as cumulative(%) normalized(normalized to total rotation) phase .The variance and phase spectra along with statistical normal distribution is shown in Figure 6. The 'goodness of fit' (statistical chisqr test)between the variance spectrum and statistical normal distribution is significant at <= 5% level. The phase spectrum is close to the statistical normal distribution, but the 'goodness of fit' is not statistically significant. However , the 'goodness of fit' between variance and phase spectra are statistically significant (chisqr test) for individual dominant wavebands (Figures 7a and 7b).
Figure 6: The variance and phase spectra along with statistical normal distribution
The cumulative percentage contribution to total variance and the cumulative(%) normalized phase(normalized w.r.t. the total rotation) for each dominant waveband is computed for significant wavebands and shown in Figures 7a and 7b to illustrate Berry's phase,namely the progressive increase in phase with increase in period and also the close association between phase and variance(see Section 2).
Figure 7a: Illustration of Berry 's phase in quantumlike chaos in prime number distribution. The phase and variance spectra are the same for prime number spacing intervals up to 10.
Figure 7b: Illustration of Berry 's phase in quantumlike chaos in prime number distribution. The phase and variance spectra are the same for prime number spacing intervals from 10 to 50.
The statistically significant(less than or equal to 5% level) wavebands are shown in Figure 8.
Figure 8: Continuous periodogram analysis results : Dominant (normalised variance greater than 1) statistically significant wavebands.
Table 1 gives the list
of a total of 110 dominant(normalised variance greater than 1) wavebands
obtained from the continuous periodogram analyses for the data set (prime
numbers in the interval 3 to 1000 at unit class intervals) . The symbol
*
indicates that the dominant waveband is statistically significant at <=
5% level. There are 14 significant dominant wavebands (Figure 8). The
dominant peak periodicities are in close agreement with model predicted
dominant peak periodicities,e.g 2.2, 3.6, 5.8, 9.5, 15.3, 24.8, 40.1,
and 64.9 prime number spacing intervals for values of n ranging
from 1 to 6 (Equation 4).The symbol S
indicates that the normalised variance and phase spectra follow each other
closely (the 'goodness of fit ' being significant at <= 5% )
displaying Berry 's phase in the quantumlike chaos exhibited by
prime number distribution . Earlier study by Marek Wolf (May 1996, IFTUWr
908/96 http://rose.ift.uni.wroc.pl/~mwolf
) also shows that the number of Twins(spacing interval 2) and primes separated
by a gap of length 4 ( "cousins") is almost the same and it determines
a fractal structure on the set of primes. The conjecture that there should
be approximately equal numbers of prime power pairs differing by 2
and by 4 , but about twice as many differing by 6
is proved to be true by Gopalkrishna Gadiyar and Padma(1999 http://www.maths.ex.ac.uk/~mwatkins/zeta/padma.pdf).
The dominant perodicities shown above at Figure 8 are consistent with these
reported results.
The period T_{50}
upto which the cumulative percentage contribution to total variance is
equal to 50 is found to be equal to 3.242
spacing interval between two adjacent primes.
This periodogram estimate of T_{50} for the prime
numbers in the interval 3 to 1000 is in
approximate agreement with model predicted value of T_{50}
approximately equal to 3.6 (Equation 9). The dominant significant
period 2 corresponds to twin primes . In number
theory [Rose,1995; Beiler,1966] the
twin prime conjecture
states that there are many pairs of primes p,q where q
= p + 2 . There are infinitely many prime pairs as
z
tends to infinity.
Table 1
















































































































































































































































































































































3.4 Spiral Pattern of Prime number distribution in the xy plane
The z^{th} prime is approximately equal to zln z [ Allenby and Redfern, 1989 ]. In the following it is shown that the prime numbers are arranged in a spiral pattern in the xy plane. The eqiangular logarithmic spiral shown at Figure 2 is given by the relation
The z^{th}
prime number has an angular phase difference equal to 1/z radians
from the earlier (z1)^{th} prime. The
spiral arrangement of the first 20 and 100 primes are shown respectively
in Figures 9 and 10. Spiral patterns in the arrangement of prime numbers
have been reported earlier by mathematicians, e.g (http://zaphod.uchicago.edu/~bryan/spiral/index.html
.)
Figure 9: The spiral pattern traced in the xy plane by the first 20 prime numbers
Figure 10: The location in the xy plane of the first 100 prime numbers. The spiralling arms closely resemble phyllotaxislike patterns such as that seen in the familiar spiral patterns found in the arrangement of leaves on a stem, in florets of composite flowers, the pattern of scales on pineapple and pine cone, etc. http://xxx.lanl.gov/abs/chaodyn/9806001
4. CONCLUSION
In mathematics Cantorian fractal
spacetime is now associated with reference to quantum systems [Nottale,1989;
Ord, 1983; El Naschie, 1993; El Naschie,1998]. Recent studies indicate
a close association between number theory in mathematics, in particular,
the distribution of prime numbers and the chaotic orbits of excited
quantum systems such as the hydrogen atom [Cipra, 1996; Berry,1992; Cipra
http://www.maths.ex.ac.uk/~mwatkins/zeta/cipra.htm
]. The cell dynamical system model presented in this paper shows that quantumlike
chaos incorporates prime number distribution functions in the description
of atmospheric flow dynamics.
Acknowledgements
The author is grateful to Dr. A. S. R. Murty for his keen interest and encouragement during the course of the study.
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