Ramanujan-Fourier series, the Wiener-Khintchine formula and the distribution of prime pairs (1999)
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BibTeX
@MISC{Gadiyar99ramanujan-fourierseries,,
author = {H. Gopalkrishna Gadiyar and R. Padma Ramanujan and R. Padma},
title = {Ramanujan-Fourier series, the Wiener-Khintchine formula and the distribution of prime pairs},
year = {1999}
}
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Abstract
The Wiener--Khintchine 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 Wiener--Khintchine 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; Ramanujan--Fourier series; Wiener--Khintchine formula 1. Introduction " The Wiener--Khintchine 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...
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