On the Asymptotic Density of Prime k-tuples and a Conjecture of Hardy and Littlewood
Grand Duchy of Luxembourg
Rue des Tanneurs 7
L-6790 Grevenmacher
E-mail: uk.laszlo.toth@gmail.com
Received:
Received: 24 July 2019; revised: 21 August 2019; accepted: 28 August 2019; published online: 16 September 2019
DOI: 10.12921/cmst.2019.0000033
Abstract:
In 1922 Hardy and Littlewood proposed a conjecture on the asymptotic density of admissible prime k-tuples. In 2011, Wolf computed the “Skewes number” for twin primes, i.e., the first prime at which a reversal of the Hardy-Littlewood inequality occurs. In this paper, we find “Skewes numbers” for 8 more prime k-tuples and provide numerical data in support of the Hardy-Littlewood conjecture. Moreover, we present several algorithms to compute such numbers.
Key words:
asymptotic density, conjecture, prime k-tuple, Skewes number
References:
[1] R.P. Brent, Irregularities in the distribution of primes and twin primes, Math. Comp. 29, 43–56 (1975).
[2] M. Wolf, The Skewes number for twin primes: counting sign changes of π2(x) – C2Li2(x), Comput. Methods Sci. Technol. 17, 87–92 (2011).
[3] M. Wolf, Random walk on the prime numbers, Physica A 250, 335–344 (1998).
[4] Th.R. Nicely, New evidence for the infinitude of some prime constellations, (2004). http://www.trnicely.net/ipc/ipc1d.pdf
[5] G.H. Hardy, J.E. Littlewood, Some problems of ’Partitio Numerorum’ III: On the expression of a number as a sum of primes, Acta Math. 44, 1–70 (1922).
[6] J.E. Littlewood, Sur la distribution des nombres premiers, C. R. Math. Acad. Sci. Paris 158, 1869–1872 (1914).
[7] S. Skewes, On the difference π(x) – li(x), J. London Math. Soc. 8, 277–283 (1933).
[8] S. Skewes, On the difference π(x) – li(x) (II), Proc. London Math. Soc. 5, 48–70 (1955).
In 1922 Hardy and Littlewood proposed a conjecture on the asymptotic density of admissible prime k-tuples. In 2011, Wolf computed the “Skewes number” for twin primes, i.e., the first prime at which a reversal of the Hardy-Littlewood inequality occurs. In this paper, we find “Skewes numbers” for 8 more prime k-tuples and provide numerical data in support of the Hardy-Littlewood conjecture. Moreover, we present several algorithms to compute such numbers.
Key words:
asymptotic density, conjecture, prime k-tuple, Skewes number
References:
[1] R.P. Brent, Irregularities in the distribution of primes and twin primes, Math. Comp. 29, 43–56 (1975).
[2] M. Wolf, The Skewes number for twin primes: counting sign changes of π2(x) – C2Li2(x), Comput. Methods Sci. Technol. 17, 87–92 (2011).
[3] M. Wolf, Random walk on the prime numbers, Physica A 250, 335–344 (1998).
[4] Th.R. Nicely, New evidence for the infinitude of some prime constellations, (2004). http://www.trnicely.net/ipc/ipc1d.pdf
[5] G.H. Hardy, J.E. Littlewood, Some problems of ’Partitio Numerorum’ III: On the expression of a number as a sum of primes, Acta Math. 44, 1–70 (1922).
[6] J.E. Littlewood, Sur la distribution des nombres premiers, C. R. Math. Acad. Sci. Paris 158, 1869–1872 (1914).
[7] S. Skewes, On the difference π(x) – li(x), J. London Math. Soc. 8, 277–283 (1933).
[8] S. Skewes, On the difference π(x) – li(x) (II), Proc. London Math. Soc. 5, 48–70 (1955).
