Continuous periodogram analyses of the

** z(s) =
1+1/2^{s} +1/3^{s} +1/4^{s} +1/5^{s} +
......... if x > 1**.

(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

In 1859

A proof of

A cell dynamical system model developed by the author shows that quantum-like chaos is inherent to

(2)

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

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

(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) ** E_{n}** in the dominant
wavebands will be given by the relation

*E _{n}=T_{S}(2+t*
)

(3)

where ** t**
is the

The model predicted periodicities (or length scales)
in terms of the primary perturbation length scale units are are ** 2.2**,

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

The overall logarithmic spiral flow structure is given by the relation

(4)

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

*1/k @
2.62*

(5)

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

In Equation 4,

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

(6)

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

(7)

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

The periodicities (or length scales)

The power spectrum, when plotted as normalised standard deviation

*T _{50} = (2+t
)t^{0
}@
3.6 unit spacing intervals*

(8)

(9)

(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} ].

*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

Five groups of data sets (

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

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

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

The '

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7a

Figure 7b

Figure 8a

Figure 8b

Figure 9a

Figure 9b

Continuous periodogram analyses of

Results of all the data sets (ranging in length from

A possible physical explanation for the observed close relationship between the

The individual fractions

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