Geodesic
Ramanujan-Fourier series and the conjecture D of Hardy and Littlewood
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 1, pp. 251-267.

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We give a heuristic proof of a conjecture of Hardy and Littlewood concerning the density of prime pairs to which twin primes and Sophie Germain primes are special cases. The method uses the Ramanujan-Fourier series for a modified von Mangoldt function and the Wiener-Khintchine theorem for arithmetical functions. The failing of the heuristic proof is due to the lack of justification of interchange of certain limits. Experimental evidence using computer calculations is provided for the plausibility of the result. We have also shown that our argument can be extended to the $m$-tuple conjecture of Hardy and Littlewood.
DOI : 10.1007/s10587-014-0098-5
Classification : 11K65, 11N05, 62M10, 62M15
Keywords: Ramanujan-Fourier series; von Mangoldt function; twin primes; Sophie Germain prime; Wiener-Khintchine theorem 
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     title = {Ramanujan-Fourier series and the conjecture {D} of {Hardy} and {Littlewood}},
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Gadiyar, H. Gopalakrishna; Padma, Ramanathan. Ramanujan-Fourier series and the conjecture D of Hardy and Littlewood. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 1, pp. 251-267. doi : 10.1007/s10587-014-0098-5. http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0098-5/

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