z(s) = 1+1/2s +1/3s +1/4s +1/5s + ......... if x > 1.
The analytic properties of the zeta function are also related to the distribution of prime numbers. It is known that there are an infinite number of prime numbers. Though the prime numbers appear to be distributed at random among the integers, the distribution follows the approximate law that the number of primes p(x) up to the integer x is equal to x/logx where log is the natural logarithm.The actual distribution of primes fluctuate on either side of the estimated value and approach closely the estimated value for large values of x.
In 1859 Bernhard Riemann gave an exact formula for the counting function p(x) , in which fluctuations about the average are related to the value of s for which z(s) =0 , s being a complex number. Based on a few numerical computations Riemann conjectured that an important set of the zeros, namely the non-trivial zeros, all have real part equal to x = 1/2 . This is the Riemann hypothesis (Keating,1990; Devlin,1997). Numerical computations done so far agree with Riemann's hypothesis. However, a theoretical proof will establish the validity of numerous results in number theory which assume that the Riemann hypothesis is true.
A proof of Riemann hypothesis will also help physicists to compute the chaotic orbits of complex atomic systems such as a hydrogen atom in a magnetic field, to the oscillations of large nuclei (Richards, 1988; Gutzwiller, 1990; Berry, 1992; Cipra, 1996; Klarreich, 2000). It is now believed that the spectrum of Riemann zeta zeros represent the energy spectrum of complex quantum systems which exhibit classical chaos.
A cell dynamical system model developed by the author shows that quantum-like chaos is inherent to fractal space-time fluctuations exhibited by dynamical systems in nature ranging from subatomic and molecular scale quantum systems to macroscale turbulent fluid flows. The model provides a unique quantification for the fractal fluctuations in terms of the statistical normal distribution. The Riemann zero spacing intervals exhibit fractal fluctuations and the power spectrum exhibits model predicted universal inverse power law form of the statistical normal distribution. The distribution of Riemann zeros therefore exhibit quantum-like chaos.
Large eddies are visualised to grow at unit length step increments at unit intervals of time, the units for length and time scale increments being respectively equal to the enclosed small eddy perturbation length scale r and the eddy circulation time scale t .
Since the large eddy is but the average of the enclosed smaller eddies, the eddy energy spectrum follows the statistical normal distribution according to the Central Limit Theorm (Ruhla, 1992). Therefore, the variance represents the probability densities. Such a result that the additive amplitudes of the eddies, when squared, represent the probabilities is an observed feature of the subatomic dynamics of quantum systems such as the electron or photon (Maddox 1988a, 1993; Rae, 1988 ). The fractal space-time fluctuations exhibited by dynamical systems are signatures of quantum-like mechanics. The cell dynamical system model provides a unique quantification for the apparently chaotic or unpredictable nature of such fractal fluctuations (Selvam and Fadnavis, 1998). The model predictions for quantum-like chaos of dynamical systems are as follows.
(a) The observed fractal fluctuations of dynamical systems are generated by an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern for the internal structure.
(b) Conventional continuous periodogram power spectral analyses of such spiral trajectories will reveal a continuum of periodicities with progressive increase in phase.
(c) The broadband power spectrum will have embedded dominant wave-bands, the bandwidth increasing with period length. The peak periods (or length scales) En in the dominant wavebands will be given by the relation
where t is the golden mean equal to (1+Ö 5)/2 [@ 1.618] and Ts , the primary perturbation length scale. Considering the most representative example of turbulent fluid flows, namely, atmospheric flows, Ghil(1994) reports that the most striking feature in climate variability on all time scales is the presence of sharp peaks superimposed on a continuous background.
The model predicted periodicities (or length scales) in terms of the primary perturbation length scale units are are 2.2, 3.6, 5.8, 9.5, 15.3, 24.8, 40.1,and 64.9 respectively for values of n ranging from -1 to 6. Peridicities close to model predicted have been reported in weather and climate variability (Burroughs 1992; Kane 1996).
(d) The ratio r/R also represents the increment dq in phase angle q (Equation 2 ). Therefore the phase angle q represents the variance. 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 (Selvam and Fadnavis, 1998). 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; Maddox 1988b; Simon et al., 1988; Anandan, 1992). The relationship of angular turning of the spiral to intensity of fluctuations is seen in the tight coiling of the hurricane spiral cloud systems.
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 and Z is the length scale ratio equal to r/R . The constant k is equal to 1/t2(@0.382) and is identified as the universal constant for deterministic chaos in fluid flows (Selvam and Fadnavis, 1998).The steady state emergence of fractal structures is therefore equal to
1/k @ 2.62
The model predicted logarithmic wind profile relationship such as Equation 4 is a long-established (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 (Hogstrom, 1985).
In Equation 4, 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 Wn+1 and Wn is given from Equation 4 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 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 4). 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
where L is the period in years and T50 is the period up to which the cumulative percentage contribution to total variance is equal to 50 and t = 0. The variable LogT50 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 fluctuations of dynamical systems, for example, meteorological parameters, when plotted as cumulative percentage contribution to total variance versus t should follow the model predicted universal spectrum (Selvam and Fadnavis, 1998, and all references therein). The literature shows many examples of pressure, wind and temperature whose shapes display a remarkable degree of universality (Canavero and Einaudi,1987).
The periodicities (or length scales) T50 and T95 up to which the cumulative percentage contribution to total variances are respectively equal to 50 and 95 are computed from model concepts as follows.
The power spectrum, when plotted as normalised standard deviation t versus cumulative percentage contribution to total variance represents the statistical normal distribution (Equation 7), i.e., the variance represents the probability density. The normalised standard deviation values t corresponding to cumulative percentage probability densities P equal to 50 and 95 respectively are equal to 0 and 2 from statistical normal distribution characteristics. Since t represents the eddy growth step n (Equation 6) the dominant periodicities (or length scales) T50 and T95 up to which the cumulative percentage contribution to total variance are respectively equal to 50 and 95 are obtained from Equation 3 for corresponding values of n equal to 0 and 2. In the present study of fractal fluctuations of spacing intervals of adjacent Riemann zeta zeros, the primary perturbation length scale Ts is equal to unit spacing interval and T50 and T95 are obtained as
T50 = (2+t )t0 @ 3.6 unit spacing intervals
(b) Riemann zeta zeros numbered
1012 + 1 through 1012 + 104 were obtained
[ Values of gamma - 267653395647, where gamma runs over the heights of the
zeros of the Riemann zeta numbered 1012 + 1 through 1012 + 104. Thus
zero # 1012 + 1 is actually
1/2 + i * 267,653,395,648.8475231278...
Values are guaranteed to be accurate only to within 10-8 ].
(c) Riemann zeta zeros
numbered 1021 + 1 through 1021 + 104 were
[ Values of gamma - 144176897509546973000, where gamma runs over the heights
of the zeros of the Riemann zeta numbered 1021 + 1 through 1021 + 104.
Thus zero # 1021 + 1 is actually
1/2 + i * 144,176,897,509,546,973,538.49806962...
Values are not guaranteed, and are probably accurate to within 10-6 ].
(d) Riemann zeta zeros numbered
1022 + 1 through 1022 + 104 were obtained
[ Values of gamma - 1370919909931995300000, where gamma runs over the heights
of the zeros of the Riemann zeta numbered 1022 + 1 through 1022 + 104.
Thus zero # 1022 + 1 is actually
1/2 + i * 1,370,919,909,931,995,308,226.68016095...
Values are not guaranteed, and are probably accurate to within 10-6 ].
tm = (log Lm / log T50)-1
The cumulative percentage
contribution to total variance, the cumulative percentage normalized phase
(normalized with respect to the total phase rotation) and the corresponding
values were computed. The power spectra were plotted as cumulative percentage
contribution to total variance versus the normalized standard deviation
as given above. The period
L is in units of number of class
intervals, unit class interval being equal to adjacent spacing interval
of zeta zeros in the present study. Periodicities up to T50
contribute up to
50% of total variance. The phase spectra were plotted
as cumulative percentage normalized (normalized to total rotation) phase
Five groups of data sets (zeros5, zeros4, zeros3, zeros1a and zeros1b) were used. Details of these five data sets are: (i) The first three data groups, namely, zeros5, zeros4, zeros3 consist of the following thirteen data sets of the same length located at same locations in the three data files zeros5, zeros4, and zeros3 respectively (1) 1 to100 (2) 1 to 500 (3) 1 to 1000 (4) 1 to 1500 (5) 1 to 2000 (6) 1 to 3000 (7) 1 to 4000 (8) 1 to 5000 (9) 5000 to 5099 (10) 5000 to 5499 (11) 5000 to 5999 (12) 5000 to 6499 (13) 1 to 9999. (ii) The data group zeros1a consists of the first twelve data sets shown above for the first three data groups at corresponding locations in data file zeros1. (iii) The data group zeros1b consists of the following eight data sets located in data file zeros1 (1) 5000 to 14999 (2) 5000 to 9999 (3) 10000 to 19999 (4) 80000 to 89999 (5) 80000 to 80099 (6) 98000 to 98049 (7) 98009 to 98049 (8) file zeros3, 5000 to 5049.
The results of power spectral analyses for all the data sets are shown in Figures 2 to 9. The variance and phase spectra along with statistical normal distributions are shown in Figures 2 and 3 for two representative data sets of Riemann zeta zero spacing intervals. Also, for these two representative data sets, the cumulative percentage contribution to total variance and the cumulative (%) normalized phase (normalized with respect to. the total rotation) for each dominant waveband is computed for significant wavebands and shown in Figures 4 and 5 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 6 shows the following: (a) details of data files (b) data series location in the data file (c) number of data values in each series (d) the value of T50 which is the length scale up to which the cumulative percentage contribution to total variance is equal to 50 (in unit spacing intervals of Riemann zeta zeros). (e) whether the variance and phase spectra follow statistical normal distribution characteristics. The length of the data sets ranged from 50 to 10,000 values.
Figures 7 to 9 give the following additional results for the same data sets grouped according to frequency of occurrence of dominant wavebands with peak periodicities in class intervals 2 - 3, 3 - 4, 4 - 6, 6 - 12, 12 - 20, 20 - 30, 30 - 50, 50 – 80. These wavebands include the model predicted (Equation 3) dominant peak periodicities (or length scales) 2.2, 3.6, 5.8, 9.5, 15.3, 24.8, 40.1, and 64.9 (in unit spacing intervals of Riemann zeta zeros) for values of n ranging from -1 to 6. Figures 7a and 7b show the percentage number of dominant wavebands. Figures 8a and 8b show the percentage number of statistically significant (less than or equal to 5% level) dominant wavebands. Figures 9a and 9b show the percentage number of dominant wavebands, which exhibit Berry's phase, namely, the variance spectrum follows closely the phase spectrum (see Section 2).
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