Geodesic
Ramanujan-Fourier series and the conjecture D of Hardy and Littlewood
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 1, pp. 251-267 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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 
@article{10_1007_s10587_014_0098_5,
     author = {Gadiyar, H. Gopalakrishna and Padma, Ramanathan},
     title = {Ramanujan-Fourier series and the conjecture {D} of {Hardy} and {Littlewood}},
     journal = {Czechoslovak Mathematical Journal},
     pages = {251--267},
     year = {2014},
     volume = {64},
     number = {1},
     doi = {10.1007/s10587-014-0098-5},
     mrnumber = {3247459},
     zbl = {06391491},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0098-5/}
}
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
SP  - 251
EP  - 267
VL  - 64
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0098-5/
DO  - 10.1007/s10587-014-0098-5
LA  - en
ID  - 10_1007_s10587_014_0098_5
ER  - 
%0 Journal Article
%A Gadiyar, H. Gopalakrishna
%A Padma, Ramanathan
%T Ramanujan-Fourier series and the conjecture D of Hardy and Littlewood
%J Czechoslovak Mathematical Journal
%D 2014
%P 251-267
%V 64
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0098-5/
%R 10.1007/s10587-014-0098-5
%G en
%F 10_1007_s10587_014_0098_5
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

[1] Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Ann. Math. 160 (2004), 781-793. | MR | Zbl

[2] Brun, V.: The sieve of Eratosthenes and the theorem of Goldbach. The Goldbach Conjecture Y. Wang Series in Pure Mathematics 4 World Scientific, Singapore (2002), 99-136. | MR

[3] Carmichael, R. D.: Expansions of arithmetical functions in infinite series. Proc. Lond. Math. Soc., II. Ser. 34 (1932), 1-26. | DOI | MR | Zbl

[4] Cojocaru, A. C., Murty, M. R.: An Introduction to Sieve Methods and their Applications. London Mathematical Society Lecture Note Series 66 Cambridge University Press, Cambridge (2006). | MR | Zbl

[5] Dickson, L. E.: History of the Theory of Numbers, Vol. I: Divisibility and Primality. Reprint of the 1919 original. Chelsea Publishing New York (1966). | MR

[6] Einstein, A.: Method for the determination of the statistical values of observations concerning quantities subject to irregular fluctuations. Arch. Sci. Phys. et Natur. 37 (1914), 254-256.

[7] Gadiyar, H. G., Padma, R.: Ramanujan-Fourier series, the Wiener-Khintchine formula and the distribution of prime pairs. Phys. A 269 (1999), 503-510. | DOI | MR

[8] Golomb, S. W.: The lambda method in prime number theory. J. Number Theory 2 (1970), 193-198. | DOI | MR | Zbl

[9] Hardy, G. H.: Goldbach's Theorem. (A lecture to the Math. Soc. of Copenhague on 6. October 1921). Mat. Tidsskr. B 1922 (1922), 1-16.

[10] Hardy, G. H.: Note on Ramanujan's trigonometrical function $c_q(n)$ and certain series of arithmetical functions. Cambr. Phil. Soc. Proc. 20 (1921), 263-271.

[11] Hardy, G. H.: Some Famous Problems of the Theory of Numbers and in Particular Waring's Problem. An inaugural lecture delivered before the University of Oxford. Clarendon Press Oxford (1920).

[12] Hardy, G. H., Littlewood, J. E.: Some problems of ``Partitio numerorum'' III: On the expression of a number as a sum of primes. Acta Math. 44 (1923), 1-70. | DOI | MR

[13] Hardy, G. H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Lond. M. S. Proc. 17 (1918), 75-115; Collected Papers of Srinivasa Ramanujan AMS Chelsea Publ., Providence (2000), 276-309. | MR

[14] Khintchine, A.: Korrelationstheorie der stationären stochastischen Prozesse. Math. Ann. 109 (1934), 604-615 German. | DOI | MR | Zbl

[15] Kittel, C.: Elementary Statistical Physics. John Wiley New York (1958). | MR

[16] Rademacher, H.: Topics in Analytic Number Theory. E. Grosswald et al. Die Grundlehren der mathematischen Wissenschaften. Band 169 Springer, New York (1973). | MR | Zbl

[17] Ramanujan, S.: On certain trigonometrical sums and their applications in the theory of numbers. Trans. Cambridge Philos. Soc. 22 (1918), 259-276; Collected Papers of Srinivasa Ramanujan AMS Chelsea Publ., Providence (2000), 179-199. | MR

[18] Stinson, D. R.: Cryptography. Theory and Practice. Series on Discrete Mathematics and its Applications CRC Press, Boca Raton (1995). | MR | Zbl

[19] Wiener, N.: Generalized harmonic analysis. Acta Math. 55 (1930), 117-258. | DOI | MR

[20] Yaglom, A. M.: Einstein's 1914 paper on the theory of irregularly fluctuating series of observations. ASSP Magazine, IEEE 4 (1987), 7-11. | DOI

[21] Zhang, Y.: Bounded gaps between primes. Annals of Math. 179 (2014), 1121-1174. | DOI | MR | Zbl

Cité par Sources :