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Ramanujan - Fourier Series and the Density of Sophie Germain Primes

H. Gopalkrishna Gadiyar·R. Padma
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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.

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H. Gopalkrishna Gadiyar, R. Padma. 2005-08-31. Ramanujan - Fourier Series and the Density of Sophie Germain Primes. https://arxiv.org/abs/math/0508639

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