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**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 co-ordinates [ 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

(1)

Constant value was assumed for the rate of change of the variable ** X_{n} ** at computational step

(2)

Numerical solutions obtained using Equation(2), which is basically a numerical integration procedure, involve iterative computations with feedback and amplification of round-off 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 long-range spatiotemporal correlations. The irregular fluctuations of real world dynamical syatems also exhibit such non-local 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 *self-organized 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 space-time structures is common to real world as well as the numerical models(Equation 2) used for simulation . A substratum of continuum fluctuations self-organizes 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 space-time 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

(3)

The dynamical evolution of space-time 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 quantum-like 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 *five-fold* 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 five-fold 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

(b) Conventional continuous periodogram power spectral analyses of such spiral trajectories will reveal a continuum of periodicities with progressive increase *d*** q** in phase angle

Figure 2: The equiangular logarithmic spiral given by (*R/r) = e***^{aq}** where

(c) The broadband power spectrum will have embedded dominant wave-bands(**R _{o}OR_{1}**,

(4)

where ** t **is the

(d) The ratio ** r/R** also represents the increment

(e) The overall logarithmic spiral flow structure is given by the relation

(5)

where the constant ** k** is the steady state fractional volume dilution of large eddy by inherent turbulent eddy fluctuations . The constant

*1/k = 2.62*

(6)

Logarithmic wind profile relationship such as Equation 5 is a long-established(observational) feature of atmospheric flows in the boundary layer, the constant ** k**, called the

In Equation 5,

statistical normalized standard deviation ** t = 0,1,2,3**, etc.

(7)

The conventional power spectrum plotted as the variance versus the frequency in log-log 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

*t = (log L / log T _{50} ) - 1*

(8)

where ** L** is the period in years and

The variable

The theoretical basis for formulation of the universal spectrum is based on the

The period ** T_{50}** up to which the cumulative percentage contribution to total variance is equal to

The power spectrum, when plotted as normalized standard deviation

*T _{50} = (2+t )t ^{0 } ~ 3.6 spacing interval between two adjacent primes*

(9)

Prime numbers with spacing intervals up to ** 3.6** or approximately

*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

Historically, the

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

In *number field* domain, the above equation can be visualized as follows. The r.m.s. circulation speeds ** W** and

Figure 3: Visualisation of the formain of large eddy (ABCD) as envelope enclosing smaller scale eddies. By analogy, the continuum number field domain(Cartesian co-ordinates ) 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 non-local connections characteristic of quantum-like 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

(10)

The steady state emergence of fractal structure ** A** is

(11)

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

from equation (5)

from equation (10)

from equation (3)

Therefore

from equation (11)

(12)

In atmospheric flows a fraction equal to ** f** of surface air is transported upward to level

The ** f** distribution represents, at level

Therefore the ratio ** P** equal to

(13)

In *number theory*, the *Prime Number Theorem* states that ** z/ln z** where

In the next Section(3.0) the following model predictions(Section 2.0) are verified.

(a) The

(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 long-range 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 '

**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

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

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

*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** (

The prime number(actual and computed) frequency distribution and also the corresponding ** f** distribution for values of

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 quantum-like 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

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 space-time fluctuations of dynamical systems can be computed accurately by an elementary, but very powerful method of analysis developed by *Jenkinson* (1977) which provides a quasi-continuous 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

*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

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 quantum-like 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 quantum-like 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

The period

Table 1

No | Periodicities in unit number class intervals | |

Peak period | Wave band | |

1 | 2.000 | 2.000 to 2.006 * |

2 | 2.010 | 2.010..to..2.010 * |

3 | 2.014 | 2.014..to..2.014 |

4 | 2.022 | 2.022..to..2.022 |

5 | 2.034 | 2.034..to..2.034 |

6 | 2.077 | 2.077..to..2.077 |

7 | 2.100 | 2.098..to..2.100 |

8 | 2.113 | 2.113..to..2.113 |

9 | 2.136 | 2.136..to..2.136 |

10 | 2.143 | 2.141..to..2.145 |

11 | 2.149 | 2.149..to..2.149 |

12 | 2.164 | 2.164..to..2.164 |

13 | 2.199 | 2.197..to..2.199 |

14 | 2.235 | 2.235..to..2.235 |

15 | 2.266 | 2.264..to..2.266 |

16 | 2.307 | 2.305..to..2.310 |

17 | 2.333 | 2.331..to..2.335 * |

18 | 2.356 | 2.356..to..2.356 |

19 | 2.364 | 2.364..to..2.366 |

20 | 2.445 | 2.443..to..2.448 |

21 | 2.467 | 2.467..to..2.470 |

22 | 2.500 | 2.497..to..2.505 * |

23 | 2.615 | 2.615..to..2.617 |

24 | 2.625 | 2.623..to..2.628 |

25 | 2.657 | 2.654..to..2.657 |

26 | 2.711 | 2.708..to..2.711 |

27 | 2.727 | 2.724..to..2.730 |

28 | 2.749 | 2.746..to..2.752 |

29 | 2.801 | 2.796..to..2.804 * |

30 | 2.890 | 2.890..to..2.890 |

31 | 2.925 | 2.925..to..2.925 |

32 | 2.936 | 2.933..to..2.936 |

33 | 2.969 | 2.969..to..2.969 |

34 | 2.999 | 2.987..to..3.014 * |

35 | 3.023 | 3.023..to..3.023 |

36 | 3.050 | 3.047..to..3.050 |

37 | 3.143 | 3.143..to..3.146 |

38 | 3.252 | 3.252..to..3.252 |

39 | 3.288 | 3.288..to..3.288 |

40 | 3.297 | 3.294..to..3.301 |

41 | 3.334 | 3.327..to..3.341 * |

42 | 3.347 | 3.347..to..3.351 |

43 | 3.361 | 3.361..to..3.364 |

44 | 3.456 | 3.442..to..3.456 |

45 | 3.470 | 3.470..to..3.473 |

46 | 3.498 | 3.491.to..3.505 *S |

47 | 3.537 | 3.537.to..3.540 |

48 | 3.558 | 3.558..to..3.558 |

49 | 3.666 | 3.663..to..3.670 |

50 | 3.688 | 3.685..to..3.688 |

51 | 3.714 | 3.707..to..3.722 |

52 | 3.748 | 3.744..to..3.755 |

53 | 3.782 | 3.778..to..3.782 |

54 | 3.820 | 3.816..to..3.823 |

55 | 3.881 | 3.881..to..3.881 |

56 | 4.125 | 4.125..to..4.125 |

57 | 4.200 | 4.196..to..4.204 |

58 | 4.247 | 4.242..to..4.247 |

59 | 4.285 | 4.281..to..4.294 |

60 | 4.333 | 4.320..to..4.346 S |

61 | 4.372 | 4.367..to..4.376 |

62 | 4.402 | 4.394..to..4.407 |

63 | 4.568 | 4.568..to..4.568 |

64 | 4.600 | 4.596..to..4.605 |

65 | 4.670 | 4.656..to..4.679 * |

66 | 4.717 | 4.717..to..4.722 |

67 | 4.750 | 4.745..to..4.760 |

68 | 4.774 | 4.769..to..4.779 |

69 | 4.939 | 4.934..to..4.944 |

70 | 4.964 | 4.964..to..4.969 |

71 | 4.999 | 4.984..to..5.014 * S |

72 | 5.034 | 5.034..to..5.034 |

73 | 5.084 | 5.074..to..5.094 |

74 | 5.161 | 5.161..to..5.161 |

75 | 5.197 | 5.192..to..5.197 |

76 | 5.250 | 5.250..to..5.250 |

77 | 5.497 | 5.491..to..5.502 |

78 | 5.813 | 5.802..to..5.819 |

79 | 5.913 | 5.907..to..5.919 |

80 | 5.949 | 5.943..to..5.960 |

81 | 6.002 | 5.972..to..6.026 *S |

82 | 6.051 | 6.044..to..6.057 |

83 | 6.087 | 6.081..to..6.087 |

84 | 6.124 | 6.124..to..6.130 |

85 | 6.279 | 6.272..to..6.285 |

86 | 6.329 | 6.323..to..6.329 |

87 | 6.496 | 6.489..to..6.502 |

88 | 6.995 | 6.974..to..7.023 * |

89 | 7.346 | 7.324..to..7.361 |

90 | 7.509 | 7.472..to..7.539 S |

91 | 7.638 | 7.615..to..7.653 |

92 | 8.102 | 8.086..to..8.119 |

93 | 8.399 | 8.366..to..8.425 |

94 | 8.501 | 8.484..to..8.526 |

95 | 9.986 | 9.926..to..10.066 * S |

96 | 10.540 | 10.508..to..10.571 |

97 | 10.981 | 10.915..to..11.036 |

98 | 12.977 | 12.912..to..13.029 S |

99 | 13.212 | 13.212..to..13.226 |

100 | 14.001 | 13.876..to..14.128 * S |

101 | 15.031 | 14.897..to..15.152 S |

102 | 15.458 | 15.412..to..15.505 |

103 | 17.050 | 16.948..to..17.153 S |

104 | 22.113 | 21.871..to..22.335 S |

105 | 28.793 | 28.450..to..29.025 S |

106 | 29.998 | 29.581..to..30.452 S |

107 | 31.223 | 31.192..to..31.254 |

108 | 42.020 | 41.353..to..42.655 S |

109 | 198.571 | 192.702..to..204.414 S |

110 | 1534.787 | 965.984..to..3402.097 S |

*3.4 Spiral Pattern of Prime number distribution in the x-y plane*

The ** z^{th}** prime is approximately equal to

Since the eddy length scale ratio

The ** z^{th}** prime number has an angular phase difference equal to 1/z radians from the earlier (

Figure 9: The spiral pattern traced in the *x-y* plane by the first 20 prime numbers

Figure 10: The location in the *x-y* plane of the first 100 prime numbers. The spiralling arms closely resemble *phyllotaxis-like* 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/chao-dyn/9806001

**4. CONCLUSION**

In mathematics Cantorian fractal space-time 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 quantum-like 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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