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

H. Gopalakrishna Gadiyar; Ramanathan Padma

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 1, page 251-267
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Gadiyar, H. Gopalakrishna, and Padma, Ramanathan. "Ramanujan-Fourier series and the conjecture D of Hardy and Littlewood." Czechoslovak Mathematical Journal 64.1 (2014): 251-267. <http://eudml.org/doc/262036>.

@article{Gadiyar2014,
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.},
author = {Gadiyar, H. Gopalakrishna, Padma, Ramanathan},
journal = {Czechoslovak Mathematical Journal},
keywords = {Ramanujan-Fourier series; von Mangoldt function; twin primes; Sophie Germain prime; Wiener-Khintchine theorem; Ramanujan-Fourier series; von Mangoldt function; twin primes; Sophie Germain prime; Wiener-Khintchine theorem},
language = {eng},
number = {1},
pages = {251-267},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Ramanujan-Fourier series and the conjecture D of Hardy and Littlewood},
url = {http://eudml.org/doc/262036},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Gadiyar, H. Gopalakrishna
AU - Padma, Ramanathan
TI - Ramanujan-Fourier series and the conjecture D of Hardy and Littlewood
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 1
SP - 251
EP - 267
AB - 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.
LA - eng
KW - Ramanujan-Fourier series; von Mangoldt function; twin primes; Sophie Germain prime; Wiener-Khintchine theorem; Ramanujan-Fourier series; von Mangoldt function; twin primes; Sophie Germain prime; Wiener-Khintchine theorem
UR - http://eudml.org/doc/262036
ER -

References

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