Ramanujan–Fourier series, the Wiener–Khintchine formula and the distribution of prime pairs

Ramanujan-Fourier series and the conjecture D of Hardy and Littlewood

“…See for example [7]. We hope our approach can be developed along more rigorous lines into a viable theory.…”

“…Another outstanding problem in number theory is regarding the Sophie Germain primes. A positive integer p is called a Sophie Germain prime if both p and 2p + 1 are primes, (2,5), (3,7), (5,11), (11,23) for example. Again one asks the question: Are there infinitely many Sophie Germain primes?…”

Ramanujan sums for signal processing of low-frequency noise

“…Ramanujan sums c q (n) are defined as the sums of the n th powers of the q th primitive roots of the unity [8], [9] …”

“…It was recently conjectured [8] that this problem of prime pairs is also related to an autocorrelation function from the Wiener-Khintchine formula…”

“…One author (M. Planat) acknowledges N. Ratier for having pointed out the paper [8] and for his help in programming. He also acknowledges R. Padma for useful comments on that topic.…”