After discovering Selvam's work on the LANL preprint server, I got in contact
and suggested that the Riemann zeta function should be involved somehow, due
to the important role of logarithmic spirals in the articles. Recall that
the logarithmic spiral is intimately linked to the Riemann zeta function,
and to the way in which the nontrivial zeta zeros relate to the local
fluctuations in the distribution of primes.
I recieved the following message on 19/09/00:
"The physical interpretation of zeta function in terms of epsilon**infinity space refers to an eddy continuum and therefore to a logarithmic spiral with the golden mean winding number. This is consistent with your earlier email to me explaining the relationship between the zeta function and the logarithmic spiral and thereby to a continuum of eddies.
The physical interpretation for the zeros of the zeta function lying on the y-axis may be visualised as follows. The bidirectional energy flow inherent to eddy circulations results in bimodal, i.e.e, formation and dissipation respectively for phenomenological form for meanifestation of energy. Therefore, the zeros of the zeta function may lie on x = 1/2 on the y-axis, for e.g., formation of clouds in updrafts (half cycle) and dissipation of clouds in adjacent downdrafts (half cycle) creating a void or clear air space between individual clouds. (Please see http://www.geocities.com/amselvam/pns97.html). Fractal structures are a natural consequence of such alternate formation/dissipation processes in a complete energy circulation cycle.
The golden mean is intrinsic to fractal structures and the spacetime
fluctuations of real world dynamical systems are fractal."
The following appeared eight years later:
A.M. Selvam, "Universal Characteristics of Fractal Fluctuations in Prime Number Distribution" (preprint 11/2008)
[Abstract:] "The frequency of occurrence of prime numbers at unit number spacing intervals exhibits selfsimilar fractal fluctuations concomitant with
inverse power law form for power spectrum generic to dynamical systems in nature such as fluid flows, stock market fluctuations, population dynamics, etc.
The physics of long-range correlations exhibited by fractals is not yet identified. A recently developed general systems theory visualises the eddy
continuum underlying fractals to result from the growth of large eddies as the integrated mean of enclosed small scale eddies, thereby generating a
hierarchy of eddy circulations, or an inter-connected network with associated long-range correlations. The model predictions are as follows: (i) The
probability distribution and power spectrum of fractals follow the same inverse power law which is a function of the golden mean. The predicted inverse
power law distribution is very close to the statistical normal distribution for fluctuations within two standard deviations from the mean of the
distribution. (ii) Fractals signify quantumlike chaos since variance spectrum represents probability density distribution, a characteristic of quantum
systems such as electron or photon. (ii) Fractal fluctuations of frequency distribution of prime numbers signify spontaneous organisation of underlying
continuum number field into the ordered pattern of the quasiperiodic Penrose tiling pattern. The model predictions are in agreement with the probability
distributions and power spectra for different sets of frequency of occurrence of prime numbers at unit number interval for successive 1000 numbers. Prime
numbers in the first 10 million numbers were used for the study."
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