A.M.Selvam
Indian Institute of Tropical Meteorology, Pune 411 008, India
(Retired) email: selvam@ip.eth.net
website: http://www.geocities.com/amselvam
Abstract
The spacing intervals
of adjacent Riemann zeta zeros(nontrivial) exhibit fractal(irregular)
fluctuations generic to dynamical systems in nature such as fluid flows,
heart beat patterns, stock market price index, etc., and are associated
with unpredictability or chaos. The power spectra of such fractal spacetime
fluctuations exhibit universal inverse power law form and signify longrange
correlations, identified as selforganized criticality . A cell
dynamical system model developed by the author for turbulent fluid
flows provides a unique quantification for the observed power spectra in
terms of the statistical normal distribution, such that the variance represents
the statistical probability densities. Such a result that the additive
amplitudes of eddies when squared, represent the statistical probabilities
is an observed feature of the subatomic dynamics of quantum systems such
as an electron or photon. Selforganized criticality is therefore
a signature of quantumlike chaos in dynamical systems. The model concepts
are applicable to all real world(observed) and computed(mathematical model)
dynamical systems.
Continuous periodogram
analyses of the fractal fluctuations of Riemann zeta zero spacing intervals
show that the power spectra follow the unique and universal inverse power
law form of the statistical normal distribution. The Riemann zeta zeros
therefore exhibit quantumlike chaos, the spacing intervals of the zeros
representing the energy(variance) level spacings of quantumlike
chaos inherent to dynamical systems in nature. The cell dynamical system
model is a general systems theory applicable to dynamical systems of
all size scales.
1. Introduction
Riemann zeta function z(s) is a function of the complex variable s and is defined as a sum over all integers( Keating, 1990)
z(s) = 1+1/2^{s} +1/3^{s} +1/4^{s} +1/5^{s} + ......... 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)
upto 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 nontrivial 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 quantumlike chaos is inherent
to fractal spacetime 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 quantumlike chaos.
2. Cell dynamical system model
As mentioned earlier(Section 1: Introduction) power spectral analyses of fractal spacetime fluctuations exhibits inverse power law form, i.e., a selfsimilar eddy continuum. The cell dynamical system model (Mary Selvam, 1990; Selvam and Fadnavis, 1998, and all references contained therein) is a general systems theory (Capra, 1996) applicable to dynamical systems of all size scales. The model shows that such an eddy continuum can be visualised as a hierarchy of successively larger scale eddies enclosing smaller scale eddies. Eddy or wave is characterised by circulation speed and radius. Large eddies of root mean square(r.m.s) circulation speed W and radius R form as envelopes enclosing small eddies of r.m.s circulation speed w_{*} and radius r such that
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 spacetime fluctuations exhibited by dynamical
systems are signatures of quantumlike 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 quantumlike 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 wavebands, the bandwidth increasing with period length. The peak periods(or length scales) E_{n} in the dominant wavebands will be given by the relation
E_{n}=T_{S}(2+t )t ^{n}
where t
is the golden mean equal to (1+Ö
5)/2 [@ 1.618]
and T_{s} , 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 . The constant k is equal to 1/t^{2}(@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 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 ( 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 W_{n+1} and W_{n} 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 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 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
T_{50} is the period up to which the cumulative
percentage contribution to total variance is equal to 50
and t = 0. The variable
LogT_{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 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) T_{50} and T_{95} 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) T_{50} and T_{95} upto 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 T_{s} is equal to unit spacing interval and T_{50} and T_{95} are obtained as
T_{50} = (2+t )t^{0 }@ 3.6 unit spacing intervals
3. Data and Analysis
Details of the Riemann
zeta zeros(nontrivial) used in the present study are given in the following:
(a) The first 100000 zeros
were obtained from:
http://www.research.att.com/~amo/zeta_tables/zeros1
(b) Riemann zeta zeros numbered
10^12 + 1 through 10^12 + 10^4 were obtained from:
http://www.research.att.com/~amo/zeta_tables/zeros3
[ Values of gamma  267653395647, where gamma runs
over the heights of the
zeros of the Riemann zeta numbered 10^12 + 1 through
10^12 + 10^4. Thus
zero # 10^12 + 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
10^21 + 1 through 10^21 + 10^4 were obtained from:
http://www.research.att.com/~amo/zeta_tables/zeros4
[ Values of gamma  144176897509546973000, where
gamma runs over the heights
of the zeros of the Riemann zeta numbered 10^21
+ 1 through 10^21 + 10^4.
Thus zero # 10^21 + 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
10^22 + 1 through 10^22 + 10^4 were obtained from:
http://www.research.att.com/~amo/zeta_tables/zeros5
[ Values of gamma  1370919909931995300000, where
gamma runs over the heights
of the zeros of the Riemann zeta numbered 10^22
+ 1 through 10^22 + 10^4.
Thus zero # 10^22 + 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) ].
3.1 Fractal structure of spacing intervals of adjacent Riemann zeta zeros
The
spacing interval between adjacent zeta zeros for a representative sample
of 100 successive zeta zeros starting from the 80,000th value are plotted
in Figure 1. The irregular zigzag pattern of fluctuations of adjacent
spacing intervals is identified as characteristic of fractal fluctuations
exhibited by dynamical systems ,such as, rainfall, river flows, stock market
price index ,etc.(Selvam and Fadnavis, 1998).
Figure 1
3.2 Continuous periodogram analyses of fractal structure of spacing intervals of adjacent Riemann zeta zeros
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 adjacent spacing intervals for different range of zeta zeros were 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 7).
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 of class intervals, unit class interval being equal to adjacent spacing interval of zeta zeros in the present study. Periodicities up to T_{50} contribute up to 50% of total variance. The phase spectra were plotted as cumulative percentage normalized (normalized to total rotation) phase .The variance and phase spectra along with statistical normal distributions are shown in Figures 2 and 3 for two representative data sets of zeta zero spacing intervals. The 'goodness of fit' (statistical chisquare 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 (chisquare test) for individual dominant wavebands (Figures 4 and 5).
Figure 2
Figure 3
Figure 4
Figure 5
3.3 Results of power spectral analyses
Continuous periodogram
analyses of the fractal fluctuations of Riemann zeta zeros was done for
a large number of data sets and the results are given in Tables1 and 2.
Table 1 lists the following:
(a) details of data files (b) data series location in the data file (c)
number of values (d) mean and standard deviation of the data series (e)
whether the data series follow statistical normal distribution (f) the
value of t_{50} which is the length scale up to which
the cumulative percentage contribution to total variance is equal to 50
(g) whether the variance and phase spectra follow statistical normal distribution
characteristics. The length of the data sets ranged from 50 to 10,000
values.
Results of power spectral
analyses of Riemann zeta zero spacing intervals agree with the following
model predictions: (a) almost all variance spectra follow statistical normal
distribution (b) The magnitude of t_{50} values
are very close to model predicted value of 3.6 unit spacing intervals(
see equation 8 ).
Results of power spectral analyses of spacing intervals of adjacent Riemann zeta zeros (nontrivial): comparision with statistical normal distribution



































































































































































































































































































































































































































































































* indicates that the data does not follow
statistical normal distribution.
t_{50}
is in terms of unit class interval (equal to the spacing interval between
adjacent zeta zeros). Periodicities upto t_{50} contribute
up to 50% to the total variance.
cvar denotes cumulative variance spectrum.
cumphs denotes cumulative phase spectrum.
cnor denotes cumulative normal distribution(statistical).
S denotes statistical significance for 'goodness
of fit' at less than or equal to 5% level.
N denotes not statistically significant for
'goodness of fit'.
Table2 gives the following
additional results for the same data sets tabulated in Table 1: (a) The
total number of dominant wavebands (b) The percentage number of dominant
wavebands with dominant 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 2) 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 zeta zeros) for values
of n ranging from 1 to 6. (c) Also listed
are , the actual number of dominant wavebands(nomtot), the number
of statistically significant wavebands(nomsg), and wavebands(nombe)
which exhibit Berry's phase , namely, the variance spectrum follows
closely the phase spectrum (see Section 2).
Table 2
Results of power spectral analyses of spacing intervals of adjacent Riemann zeta zeros (nontrivial): dominant wavebands


















nomtot nomsg nombe 
6 3 3 
3 0 2 
2 0 2 
2 0 1 
1 0 1 




nomtot nomsg nombe 
33 8 7 
12 4 6 
12 2 5 
7 1 4 




nomtot nomsg nombe 
61 21 4 
33 5 2 
26 3 9 
16 2 4 
7 0 2 
2 0 2 




nomtot nomsg nombe 
83 24 0 
53 10 0 
40 8 2 
26 1 2 
9 0 4 
2 0 2 
1 0 1 




nomtot nomsg nombe 
91 32 0 
68 10 0 
52 6 0 
38 1 1 
11 0 2 
2 0 2 
1 0 1 
1 0 1 



nomtot nomsg nombe 
102 49 0 
69 26 0 
67 6 0 
60 1 0 
11 0 2 
3 0 2 
1 0 1 
1 0 1 



nomtot nomsg nombe 
102 60 0 
72 26 0 
86 10 0 
85 1 0 
12 5 2 
3 2 2 
2 0 1 
1 0 1 



nomtot nomsg nombe 
107 63 0 
78 28 0 
86 14 0 
103 1 0 
13 8 0 
3 2 1 
2 1 1 
1 0 1 



nomtot nomsg nombe 
6 1 3 
1 0 1 
3 0 1 
4 0 3 




nomtot nomsg nombe 
31 8 5 
13 2 4 
16 3 7 
13 0 7 
1 0 0 




nomtot nomsg nombe 
69 21 1 
31 4 2 
31 4 4 
21 0 5 
6 0 2 
2 0 2 
1 0 1 




nomtot nomsg nombe 
87 26 0 
48 7 0 
48 6 1 
38 2 4 
9 0 5 
2 0 2 
1 0 1 
1 0 1 



nomtot nomsg nombe 
5 2 4 
5 1 5 
5 0 4 
2 0 2 




nomtot nomsg nombe 
37 8 5 
16 2 7 
20 1 10 
9 0 4 




nomtot nomsg nombe 
69 14 0 
30 6 1 
38 3 3 
26 0 6 
9 0 4 
2 0 2 
1 0 0 




nomtot nomsg nombe 
91 23 0 
41 10 2 
51 4 0 
35 0 2 
9 0 4 
2 0 2 
1 0 1 




nomtot nomsg nombe 
94 33 0 
57 17 1 
60 6 0 
51 0 0 
9 0 5 
2 0 2 
1 0 1 
1 0 1 



nomtot nomsg nombe 
104 46 0 
64 19 0 
95 12 0 
74 1 0 
9 2 4 
3 2 2 
1 0 1 
1 0 1 



nomtot nomsg nombe 
104 48 0 
69 32 0 
99 14 0 
93 2 0 
9 9 1 
3 2 2 
2 1 1 
1 0 1 



nomtot nomsg nombe 
103 52 0 
67 37 0 
90 19 0 
109 4 0 
10 9 1 
3 2 0 
2 1 1 
1 0 1 



nomtot nomsg nombe 
6 3 4 
2 0 1 
2 0 0 




nomtot nomsg nombe 
35 12 7 
18 2 3 
11 0 6 
12 0 6 
1 0 0 




nomtot nomsg nombe 
58 14 3 
35 4 4 
30 1 0 
17 0 4 
5 0 2 
2 0 2 
1 0 1 




nomtot nomsg nombe 
81 24 0 
52 8 0 
45 2 1 
24 0 2 
9 0 6 
2 0 2 
1 0 1 
1 0 1 



nomtot nomsg nombe 
8 1 5 
3 1 3 
3 1 2 
2 0 2 




nomtot nomsg nombe 
40 9 3 
19 4 7 
17 0 5 
9 0 8 
2 0 2 
1 0 1 
1 0 1 




nomtot nomsg nombe 
61 17 1 
32 8 3 
36 0 0 
13 4 10 
3 2 3 
1 0 1 
1 0 1 




nomtot nomsg nombe 
87 24 0 
46 11 1 
50 1 1 
14 9 6 
3 2 3 
1 1 1 
1 0 1 




nomtot nomsg nombe 
100 32 0 
59 13 1 
52 3 0 
18 13 3 
3 2 2 
1 1 1 
1 1 1 




nomtot nomsg nombe 
111 45 0 
69 19 0 
69 22 0 
19 14 0 
3 2 1 
1 1 1 
1 1 1 




nomtot nomsg nombe 
96 51 0 
63 25 0 
66 36 0 
19 15 0 
4 3 1 
1 1 1 
1 1 1 




nomtot nomsg nombe 
103 61 0 
71 32 0 
64 44 0 
21 15 0 
5 3 0 
1 1 1 
1 1 1 




nomtot nomsg nombe 
2 1 1 
1 1 0 
2 0 1 
1 0 1 




nomtot nomsg nombe 
7 1 4 
4 1 4 
5 0 5 
3 0 2 
2 0 2 




nomtot nomsg nombe 
41 8 4 
16 2 3 
16 2 8 
13 0 8 
2 0 2 
1 0 1 
1 0 1 




nomtot nomsg nombe 
66 20 1 
30 7 3 
35 4 4 
16 2 4 
3 2 2 
1 0 1 
1 0 1 




nomtot nomsg nombe 
84 28 1 
41 7 0 
40 6 1 
20 9 4 
3 2 3 
1 1 1 
1 0 1 




nomtot nomsg nombe 
8 0 5 
3 0 3 
3 0 3 
0 0 0 
0 0 0 
2 0 2 
0 0 0 
0 0 0 



nomtot nomsg nombe 
36 4 2 
17 0 3 
15 0 2 
7 0 2 
0 0 0 
0 0 0 
4 0 2 
0 0 0 



nomtot nomsg nombe 
72 6 0 
38 0 0 
22 0 0 
12 0 0 
0 0 0 
0 0 0 
2 0 2 
3 0 1 



nomtot nomsg nombe 
107 15 0 
49 1 0 
37 0 0 
20 0 0 
0 0 0 
0 0 0 
0 0 0 
6 0 2 



nomtot nomsg nombe 
113 17 0 
65 0 0 
62 0 0 
23 0 0 
0 0 0 
0 0 0 
0 0 0 
8 0 2 



nomtot nomsg nombe 
109 30 0 
87 7 0 
83 0 0 
28 0 0 
0 0 0 
0 0 0 
0 0 0 
6 0 0 



nomtot nomsg nombe 
109 36 0 
87 7 0 
92 1 0 
34 0 0 
0 0 0 
0 0 0 
0 0 0 
0 0 0 



nomtot nomsg nombe 
101 54 0 
82 12 0 
90 3 0 
45 1 0 
0 0 0 
0 0 0 
0 0 0 
0 0 0 



nomtot nomsg nombe 
9 3 4 
2 1 1 
2 1 1 
2 1 1 




nomtot nomsg nombe 
29 8 1 
7 3 0 
3 2 1 
4 2 0 




nomtot nomsg nombe 
53 9 0 
5 2 0 
2 2 0 
2 2 0 




nomtot nomsg nombe 
64 19 0 
13 2 0 
2 2 0 
2 2 0 




nomtot nomsg nombe 
100 77 0 
60 32 0 
11 3 0 
3 3 0 
0 0 0 
0 0 0 
0 0 0 
0 0 0 



nomtot nomsg nombe 
103 61 0 
33 10 1 
20 1 0 
2 2 0 
0 0 0 
0 0 0 
0 0 0 
0 0 0 



nomtot nomsg nombe 
93 65 0 
41 29 0 
19 3 1 
2 2 1 
0 0 0 
0 0 0 
0 0 0 
0 0 0 



nomtot nomsg nombe 
80 49 0 
10 6 1 
5 4 0 
2 2 0 
1 1 0 




nomtot nomsg nombe 
9 2 7 
6 0 6 
4 1 4 
1 0 1 
1 0 1 




nomtot nomsg nombe 
3 2 1 
2 0 2 
3 0 3 
1 0 1 




nomtot nomsg nombe 
5 1 2 
2 0 0 
2 0 2 
1 0 1 
1 0 1 




nomtot nomsg nombe 
3 2 1 
2 0 2 
3 0 3 
1 0 1 




nomtot nomsg nombe 
113 79 0 
69 43 0 
94 36 0 
113 33 0 
14 11 0 
4 2 0 
2 1 0 
1 1 1 



nomtot nomsg nombe 
95 77 0 
69 42 0 
101 33 0 
106 32 0 
11 9 0 
4 2 0 
2 1 0 
1 1 0 



nomtot nomsg nombe 
99 81 0 
73 51 0 
56 43 0 
19 17 0 
4 3 0 
3 1 0 
3 1 0 
unit class interval is equal to the spacing between adjacent zeta zeros(nontrivial).
nomtot denotes the total number of dominant wavebands in each class interval.
nomsg denotes the number of statistically significant (<= 5%) wavebands in each class interval.
nombe denotes the number of wavebands where the variance and phase spectra are the same, a manifestation of Berry's phase in quantum systems.
4. Discussions and Conclusions
The spacing intervals
of adjacent Riemann zeta zeros(nontrivial) exhibit fractal fluctuations
ubiquitous to dynamical systems in nature. Fractal fluctuations are irregular
or chaotic and has emerged (since 1980s) as a subject of intensive study
in the new multidisciplinary science of Nonlinear Dynamics and Chaos
(Gleick, 1987; Gutzwiller,1990; Jurgen et al ., 1990; Bassingthwaighte
and Beyer,1991; Deering and West 1992; Stewart,1998). Power spectra of
fractal fluctuations exhibit inverse power law form indicating longrange
spacetime correlations identified as selforganized criticality
(Bak et al ., 1987;1988; Bak and Chen, 1989,1991; Goldberger et
al., 1990; Schroeder, 1991; Stanley, 1995; Ghashghaie et al
., 1996; Buchanan, 1997; Newman, 2000). Also, inverse power law form for
power spectra indicate that an eddy continuum underlies the apparently
irregular(or chaotic) fractal fluctuations, i.e., the superimposition of
an ensemble of eddies( say, such as sine waves) generates the observed
fractal fluctuations. A cell dynamical system model developed by the author
provides unique quantification for the power spectra of fractal fluctuations
in terms of the statistical normal distribution such that the variance
represents the probabilities. In summary, fractal fluctuations imply quantumlike
chaos in dynamical systems for the following reasons: (a) The superimposition
of an ensemble of eddies or waves results in the observed fluctuation pattern.
(b) The additive amplitudes of the eddies when squared gives the variance
which represents the probability densities. Fractal fluctuations therefore
exhibit quantumlike chaos in macroscale dynamical systems.
Continuous periodogram
analyses of Riemann zeta zero spacing intervals show that the power spectra
follow the universal and unique inverse power law form of the statistical
normal distribution. Riemann zeta zero spacing intervals therefore exhibit
quantumlike chaos and is consistent with similar studies by the author
which have shown that prime number distribution also exhibits quantumlike
chaos ( Selvam and Fadnavis,2000; Selvam, 2000). Riemann had shown that
the zeta function represents prime number distribution. Observational and
computed values of energy level distributions of excited quantum systems
appear to follow closely the Riemann zeta zeros and also prime number distribution(Cipra,
1996). The results are consistent with cell dynamical system model prediction
that fractal fluctuations are signatures of quantumlike chaos in dynamical
systems of all sizes ranging from the subatomic quantum systems to macroscale
fluid flows. The Heisenberg uncertainty principle for quantum systems implies
unpredictable fluctuations, i.e., fractal spacetime fluctuations (Hey
and Walters, 1989) which is a signature of quantumlike chaos.
Results of all the
data sets(ranging in length from 50 to 10000
values) show that starting from the high frequency side, periodicities
upto model predicted value of about 3.6 unit spacing intervals
contribute upto 50% to the total variance.
A possible physical
explanation for the observed close relationship between the Riemann zeta
zeros and energy levels of quantum systems is given in the following:
The individual fractions
1/2,
1/3,
1/4,
1/5,
etc., in the expression for the zeta function (Equation 1 ) may represent
(a) the probabilities of occurrence of the primary perturbation in successive
growth stages in unit length steps of the large eddy and also the length
scale ratio of the enclosed primary eddy to the large eddy . As shown
in Equation 2 , this length scale ratio (r/R) represents
the variance or eddy energy. Graphically , in the x  y plane (complex
plane) , the above fractions raised to the power of the complex number
s(=x+iy)
represent fractional probabilities corresponding to the phase angle represented
by the location coordinates x and y (Argand diagram). Therefore the zeta
function represents the energy spectrum of quantum systems at any location
(x,y). The zeta zeros on the yaxis at x=1/2, therefore represent the eddy
energy minima. A rotation by 90 degrees of these zeta zero locations
will give the energy(maximum) spectrum of the quantum system. An eddy or
wave circulation is bidirectional by concept and is associated with bimodal,
namely formation and dissipation respectively of phenomenological form
for manifestation of energy (Mary Selvam, 1990). Since manifestation of
energy in phenomenological form occurs only in onehalf cycle, the corresponding
energy levels occur at x=1/2.
5. Acknowledgement
The author is grateful
to Dr.A.S.R.Murty for his keen interest and encouragement during the course
of this study.
References
Anandan, J., 1992: The geometric phase. Nature 360, 307313.
Bak, P., Tang, C.,and Wiesenfeld K., 1987: Selforganized criticality : an explanation of 1/f noise. Phys. Rev. Lett. 59, 381384.
Bak, P.C., Tang, C., and Wiesenfeld, K., 1988: Selforganized criticality. Phys. Rev. A. 38, 364  374.
Bak, P., Chen, K., 1989: The physics of fractals. Physica D 38, 512.
Bak, P., Chen, K., 1991: Selforganized criticality. Sci. Am., January, 2633.
Bassingthwaighte, J. B. and Beyer, R. P.,1991: Fractal correlations in heterogeneous systems. Physica D 53, 7184.
Berry, M. V.,1988: The geometric phase. Sci. Amer. Dec., 2632.
Berry, M., 1992: Quantum physics on the edge of chaos. In The New Scientist's Guide to Chaos, (Ed) Nina Hall, Penguin Books, pp.184  195.
Buchanan, M., 1997: One law to rule them all. New Scientist 8 Nov., 3035.
Burroughs, W. J.,1992: Weather Cycles: Real or Imaginary? Cambridge University Press,Cambridge.
Canavero,F.G., Einaudi, F.,1987: Time and space variability of atmospheric processes. J. Atmos. Sci. 44(12),15891604.
Capra, F., 1996: The web of life , HarperCollins Publishers, London, pp.311.
Cipra, B., 1996: Prime formula weds number theory and quantum physics. Science 274, 20142015.
Deering, W., and West, B. J., 1992: Fractal physiology. IEEE Engineering in Medicine and Biology June, 4046.
Devlin, K., 1997: Mathematics: The Science of Patterns. Scientific American Library, New York, pp.215.
Ghashghaie, S., Breymann, Peinke, J., Talkner, P., Dodge, Y., 1996: Turbulent cascades in foreign exchange markets. Nature 381, 767770.
Ghil, M.,1994: Cryothermodynamics : the chaotic dynamics of paleoclimate. Physica D 77,130159.
Gleick, J., 1987: Chaos : Making a New Science. Viking , New York.
Goldberger, A. L., Rigney, D. R., West, B. J., 1990: Chaos and fractals in human physiology. Sci. Am. 262(2), 4249.
Gutzwiller, M. C., 1990: Chaos in classical and quantum systems. Interdisciplinary Applied Mathematics, Volume 1, SpringerVerlag, New York, pp.427.
Hey,T. and Walters, P., 1989: The quantum universe. Cambridge university Press, pp.180.
Hogstrom, U., 1985: Von Karman's constant in atmospheric boundary layer now reevaluated. J. Atmos. Sci. 42, 263270.
Jenkinson, A. F.,1977: A Powerful Elementary Method of Spectral Analysis for use with Monthly,Seasonal or Annual Meteorological Time Series. Meteorological Office, London, Branch Memorandum No. 57, pp. 123.
Jurgen, H., Peitgen, HO., Saupe, D.1990: The language of fractals. Sci. Amer. 263, 4049.
Keating, J., 1990: Physics and the queen of mathematics. Physics World April, 4650.
Klarreich, E., 2000: Prime time. New Scientist 11 November, 32 36.
Maddox, J., 1988a: Licence to slang Copenhagen ? Nature 332, 581.
Maddox, J., 1988b: Turning phases into frequencies. Nature 334 , 99.
Maddox, J., 1993: Can quantum theory be understood ? Nature 361, 493.
Mary Selvam, A.,1990: Deterministic chaos, fractals and quantumlike mechanics in atmospheric flows. Can. J. Phys. 68, 831841. http://xxx.lanl.gov/html/physics/0010046
Newman, M., 2000: The power of design. Nature 405, 412413.
Rae, A., 1988: Quantumphysics: illusion or reality ? Cambridge University Press, New York, p.129.
Richards, D., 1988: Order and chaos in strong fields. Nature 336, 518519.
Ruhla, C., 1992: The physics of chance. Oxford University Press, Oxford, U. K.,pp.217.
Schroeder, M., 1991: Fractals, Chaos and Powerlaws. W.H.Freeman and Co., N.Y.
Selvam, A. M., and Fadnavis, S., 1998:
Signatures of a universal spectrum for atmospheric interannual variability
in some disparate climatic regimes. Meteorology and Atmospheric Physics
66, 87112.
http://xxx.lanl.gov/abs/chaodyn/9805028
Selvam, A. M., and Fadnavis, S., 2000:
Cantorian fractal patterns, quantumlike chaos and prime numbers in atmospheric
flows.(to be submitted for Journal publication).
http://xxx.lanl.gov/abs/chaodyn/9810011
Selvam, A. M., 2000: Quantumlike chaos
in prime number distribution and in turbulent fluid flows. (to be submitted
for Journal publication).
http://xxx.lanl.gov/html/physics/0005067
Simon, R., Kimble, H. J., Sudarshan, E.C.G., 1988: Evolving geometric phase and its dynamical interpretation as a frequency shift: an optical experiment. Phys.Rev.Letts. 61(1),1922.
Stanley, H.E., 1995: Powerlaws and universality. Nature 378, 554.
Stewart, I., 1998: Life's other secret
. Allen Lane, The penguin press, pp.273.