Ramanujan - Fourier Series and the Density of Sophie Germain Primes

TL;DR

This paper introduces a heuristic approach using Ramanujan-Fourier series to estimate the density of Sophie Germain primes, supported by computational evidence, offering a new perspective on their distribution.

Contribution

It presents a novel heuristic method employing Ramanujan-Fourier series to approximate the density of Sophie Germain primes in a generalized framework.

Findings

Heuristic formula for the density of Sophie Germain primes.

Use of Ramanujan-Fourier series for prime density estimation.

Supporting computational evidence for the proposed formula.

Abstract

A prime p is called Sophie Germain prime if 2p+1 is also prime. A formula for the density of such primes is given in a more general setting using a new approach. This method uses the Ramanujan-Fourier series for a modified von Mangoldt function. The proof remains heuristic as interchange of certain limits has not been justified. Experimental evidence using computer calculations is provided for the plausibility of the result.