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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...
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