Rota meets Ramanujan: Probabilistic interpretation of Ramanujan - Fourier series

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 Wiener-Khintchine 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 one due to Rota which is combinatorial and cast in the modern, abstract 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) = ∞ ∑