| Title:
|
Ramanujan-Fourier series and the conjecture D of Hardy and Littlewood (English) |
| Author:
|
Gadiyar, H. Gopalakrishna |
| Author:
|
Padma, Ramanathan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
1 |
| Year:
|
2014 |
| Pages:
|
251-267 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Ramanujan-Fourier series |
| Keyword:
|
von Mangoldt function |
| Keyword:
|
twin primes |
| Keyword:
|
Sophie Germain prime |
| Keyword:
|
Wiener-Khintchine theorem |
| MSC:
|
11K65 |
| MSC:
|
11N05 |
| MSC:
|
62M10 |
| MSC:
|
62M15 |
| idZBL:
|
Zbl 06391491 |
| idMR:
|
MR3247459 |
| DOI:
|
10.1007/s10587-014-0098-5 |
| . |
| Date available:
|
2014-09-29T10:04:31Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143964 |
| . |
| Reference:
|
[1] Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P.Ann. Math. 160 (2004), 781-793. Zbl 1071.11070, MR 2123939 |
| Reference:
|
[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 0677199 |
| Reference:
|
[3] Carmichael, R. D.: Expansions of arithmetical functions in infinite series.Proc. Lond. Math. Soc., II. Ser. 34 (1932), 1-26. Zbl 0004.29305, MR 1576142, 10.1112/plms/s2-34.1.1 |
| Reference:
|
[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). Zbl 1121.11063, MR 2200366 |
| Reference:
|
[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 0245499 |
| Reference:
|
[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. |
| Reference:
|
[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. MR 1702866, 10.1016/S0378-4371(99)00171-5 |
| Reference:
|
[8] Golomb, S. W.: The lambda method in prime number theory.J. Number Theory 2 (1970), 193-198. Zbl 0198.37601, MR 0257013, 10.1016/0022-314X(70)90019-3 |
| Reference:
|
[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. |
| Reference:
|
[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. |
| Reference:
|
[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). |
| Reference:
|
[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. MR 1555183, 10.1007/BF02403921 |
| Reference:
|
[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 2280879 |
| Reference:
|
[14] Khintchine, A.: Korrelationstheorie der stationären stochastischen Prozesse.Math. Ann. 109 (1934), 604-615 German. Zbl 0008.36806, MR 1512911, 10.1007/BF01449156 |
| Reference:
|
[15] Kittel, C.: Elementary Statistical Physics.John Wiley New York (1958). MR 0096402 |
| Reference:
|
[16] Rademacher, H.: Topics in Analytic Number Theory.E. Grosswald et al. Die Grundlehren der mathematischen Wissenschaften. Band 169 Springer, New York (1973). Zbl 0253.10002, MR 0364103 |
| Reference:
|
[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 2280864 |
| Reference:
|
[18] Stinson, D. R.: Cryptography. Theory and Practice.Series on Discrete Mathematics and its Applications CRC Press, Boca Raton (1995). Zbl 0855.94001, MR 1327060 |
| Reference:
|
[19] Wiener, N.: Generalized harmonic analysis.Acta Math. 55 (1930), 117-258. MR 1555316, 10.1007/BF02546511 |
| Reference:
|
[20] Yaglom, A. M.: Einstein's 1914 paper on the theory of irregularly fluctuating series of observations.ASSP Magazine, IEEE 4 (1987), 7-11. 10.1109/MASSP.1987.1165596 |
| Reference:
|
[21] Zhang, Y.: Bounded gaps between primes.Annals of Math. 179 (2014), 1121-1174. Zbl 1290.11128, MR 3171761, 10.4007/annals.2014.179.3.7 |
| . |
