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

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Abstract

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.

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Authors and Affiliations

  1. School of Advanced Sciences, V.I.T. University, Vellore, 632014, India

    H. Gopalakrishna Gadiyar & Ramanathan Padma

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  1. H. Gopalakrishna Gadiyar
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  2. Ramanathan Padma
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Correspondence to H. Gopalakrishna Gadiyar.

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Gopalakrishna Gadiyar, H., Padma, R. Ramanujan-Fourier series and the conjecture D of Hardy and Littlewood. Czech Math J 64, 251–267 (2014). https://doi.org/10.1007/s10587-014-0098-5

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  • DOI: https://doi.org/10.1007/s10587-014-0098-5

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