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
En=TS(2+t )tn
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
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 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
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
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
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
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
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 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).
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7a
Figure 7b
Figure 8a
Figure 8b
Figure 9a
Figure 9b
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