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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
RamanujanFourier series, the WienerKhintchine formula and the distribution of prime pairs
, 1999
"... The WienerKhintchine formula plays a central role in statistical mechanics. It is shown here that the problem of prime pairs is related to autocorrelation and hence to a WienerKhintchine formula. "Experimental" evidence is given for this. c # 1999 Elsevier Science B.V. All rights res ..."
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The WienerKhintchine formula plays a central role in statistical mechanics. It is shown here that the problem of prime pairs is related to autocorrelation and hence to a WienerKhintchine formula. "Experimental" evidence is given for this. c # 1999 Elsevier Science B.V. All rights reserved. PACS: 05.40+j; 02.30.Nw; 02.10.Lh Keywords: Twin primes; RamanujanFourier series; WienerKhintchine formula 1. Introduction " The WienerKhintchine theorem states a relationship between two important characteristics of a random process: the power spectrum of the process and the correlation function of the process" [1]. One of the outstanding problems in number theory is the problem of prime pairs which asks how primes of the form p and p+h (where h is an even integer) are distributed. One immediately notes that this is a problem of #nding correlation between primes. We make two key observations. First of all there is an arithmetical function (a function de#ned on integers) which traps the...
2008 Convolution and crosscorrelation of RamanujanFourier series Preprint 0805.0284 [math.NT
"... Abstract. One of the remarkable achievements of Ramanujan[1], Hardy[2] and Carmichael[3] was the development of RamanujanFourier series which converge to an arithmetic function. The RamanujanFourier series, for an arithmetic function, a(n), is given by ∞X a(n) = aqcq(n) q=1 where the Ramanujan su ..."
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Abstract. One of the remarkable achievements of Ramanujan[1], Hardy[2] and Carmichael[3] was the development of RamanujanFourier series which converge to an arithmetic function. The RamanujanFourier series, for an arithmetic function, a(n), is given by ∞X a(n) = aqcq(n) q=1 where the Ramanujan sum, cq(n), is defined as qX cq(n) = k=1 (k,q)=1 e 2πin k q and (k, q) is the greatest common divisor of k and q. The object of this paper is to show that if two, arithmetic functions, a(n) and b(n), have RamanujanFourier series of, ∞X ∞X a(n) = aqcq(n) and b(n) = bqcq(n) q=1 then a crosscorrelation between a(n) and b(n) can be given based on the RamanujanFourier coefficients. Specifically: 1 (1) lim N→ ∞ N n=1 q=1 NX a(n + m) ¯ ∞X b(n) = aq¯bqcq(m) where ¯x is the complex conjugate of x This paper uses the machinery of almost periodic functions to prove that even without uniform convergence the connection between a pair of almost periodic functions and the constants of the associated Fourier series exists for both the convolution and crosscorrelation. The general results for two almost periodic functions are narrowed and applied to Ramanujan sums and finally applied to support the specific relation (1). The WienerKhinchin fomula, (1), connecting the autocorrelation of an arithmetic function and the coefficients of its Ramanujan Fourier series is a powerful link between the circle method and the sieve methods found in number theory. The application of this WeinerKhinchin formula to number theory is described in the works of H. G. Gadiyar and R. Padma [4], [5], [6], [7]. The WienerKhinchin fomula is used in [4] to prove the HardyLittlewood conjecture and is used in [6] to prove the density of Sophie primes. q=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
Renormalisation and the Density of Prime Pairs
, 1998
"... Ideas from physics are used to show that the prime pairs have the density conjectured by Hardy and Littlewood. The proof involves dealing with infinities like in quantum field theory. Keywords: twin primes, Poisson summation formula, Ramanujan  Fourier expansion, renormalisation 1 Email: padma@im ..."
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Ideas from physics are used to show that the prime pairs have the density conjectured by Hardy and Littlewood. The proof involves dealing with infinities like in quantum field theory. Keywords: twin primes, Poisson summation formula, Ramanujan  Fourier expansion, renormalisation 1 Email: padma@imsc.ernet.in 2 Email: padma@imsc.ernet.in 1 This article may be considered as an invitation to theoretical physicists to enter the field of additive number theory. For sometime now there have been serious attempts to cross fertilise the disciplines of physics and number theory. It seems strange that on the one hand the most practical of disciplines, namely, physics has connections with the most aracane of disciplines, namely, number theory. However, surprising connections have appeared between number theory and physics as can be seen in [1], [3] and [9]. The work of Ramanujan in particular has had surprising connections with string theory, conformal field theory and statistical physics. For sometime now the authors one of whom is a theoretical physicist and the other a number theorist have been trying to understand problems in additive number theory using ideas from both fields. One such problem is the distribution of prime pairs. Prime pairs are numbers which are primes differing by some even integer. For example, 3; 5; 5; 7; 11; 13; 17; 19 and so on are all prime pairs with common difference 2. The question is whether such prime pairs are infinite, if so, what is the density? We will now summarise the standard method used to attack the problem which is the circle method. For technical reasons, the von Mangoldt function (n) (which is defined to be log p if n = p m where p is a prime and 0 otherwise) is used instead of the characteristic function on the primes. Hence in...