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Quantumlike Chaos in Prime Number Distribution and in Turbulent Fluid Flows
 APEIRON
, 2001
"... re applied to derive the following results for the observed association between prime number distribution and quantumlike chaos. (i) Number theoretical concepts are intrinsically related to the quantitative description of dynamical systems. (ii) Continuous periodogram analyses of different set ..."
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re applied to derive the following results for the observed association between prime number distribution and quantumlike chaos. (i) Number theoretical concepts are intrinsically related to the quantitative description of dynamical systems. (ii) Continuous periodogram analyses of different sets of adjacent prime number spacing intervals show that the power spectra follow the model predicted universal inverse powerlaw form of the statistical normal distribution. The prime number distribution therefore exhibits selforganized criticality, which is a signature of quantumlike chaos. (iii) The continuum real number field contains unique structures, namely, prime numbers, which are analogous to the dominant eddies in the eddy continuum in turbulent fluid flows. Keywords: quantumlike chaos in prime numbers, fractal structure of primes, quantification of prime number distribution, prime numbers and fluid flows 1. Introduction he continuum real number field (infinite numbe
Modular functions and Ramanujan sums for the analysis of 1/f noise in electronic circuits
, 2003
"... Abstract: A number theoretical model of 1/f noise found in phase locked loops is developed. Oscillators from the input of the non linear and low pass filtering stage are shown to lock their frequencies from continued fraction expansions of their frequency ratio, and to lock their phases from modula ..."
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Abstract: A number theoretical model of 1/f noise found in phase locked loops is developed. Oscillators from the input of the non linear and low pass filtering stage are shown to lock their frequencies from continued fraction expansions of their frequency ratio, and to lock their phases from modular functions found in the hyperbolic geometry of the half plane. A cornerstone of the analysis is the Ramanujan sums expansion of arithmetical functions found in prime number theory, and their link to Riemann hypothesis. KeyWords: Electronic circuits, number theory, signal processing, 1/f noise 1
Rota meets Ramanujan: Probabilistic interpretation of Ramanujan  Fourier series arXiv:mathph/0209066v1 [math.NT
"... In this paper the ideas of Rota and Ramanujan are shown to be central to understanding problems in additive number theory. The circle and sieve methods are two different facets of the same theme of interplay between probability and Fourier series used to great advantage by Wiener in engineering. Nor ..."
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In this paper the ideas of Rota and Ramanujan are shown to be central to understanding problems in additive number theory. The circle and sieve methods are two different facets of the same theme of interplay between probability and Fourier series used to great advantage by Wiener in engineering. Norbert Wiener forged a powerful tool for electrical engineers by combining two distinct branches of probability and Fourier series. Recently the authors in [1] have shown that the twin prime problem is related to the WienerKhintchine formula for Ramanujan Fourier expansion for a relative of the von Mangoldt function. Planat[2] has extensively developed applications of Ramanujan Fourier series in practical settings. The next natural question is: Is there a probabilistic interpretation of Ramanujan Fourier series? To our surprise we found that there are two distinct streams of thought language of characters and the other is the historically older argument of Ramanujan in the classical, concrete language of Fourier series. We would like to give the punch line right away and then give a brief summary of the view points of Rota and Ramanujan. Rota considers the group C ∞ of rational numbers modulo 1 and crucially bases his arguments summarized in the next section on C ∗ ∞ the group of characters of C∞. Ramanujan Fourier series are a(n) = aqcq(n), q=1 1 where cq(n) = q∑ k=1
LINKING THE CIRCLE AND THE SIEVE: RAMANUJAN FOURIER SERIES
, 2006
"... Currently the circle and the sieve methods are the key tools in analytic number theory. In this paper the unifying theme of the two methods is shown to be Ramanujan Fourier series. 1 Introduction. The two well known methods in additive number theory are the circle method and the sieve method. The c ..."
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Currently the circle and the sieve methods are the key tools in analytic number theory. In this paper the unifying theme of the two methods is shown to be Ramanujan Fourier series. 1 Introduction. The two well known methods in additive number theory are the circle method and the sieve method. The circle method is based on using a generating function (See Section 3) and noting along with Ramanujan and Hardy that the rational points on the circle contribute most and then through estimates showing that the contribution from the other points is