On the Asymptotic Density of Prime k-tuples and a Conjecture of Hardy and Littlewood

• Autorzy: Tóth L.

Identyfikatory

DOI

10.12921/cmst.2019.0000033

• Języki publikacji: EN

Abstrakty

EN

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 HardyLittlewood 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.

Słowa kluczowe

EN

prime k-tuple; asymptotic density; conjecture; Skewes number

• Wydawca: Institute of Bioorganic Chemistry Scientific Publishers OWN, Polish Academy of Sciences
• Czasopismo: Computational Methods in Science and Technology
• Rocznik: 2019
• Tom: Vol. 25, No. 3
• Strony: 143--148
• Opis fizyczny: Bibliogr. 8 poz., rys.

Twórcy

autor

Tóth L.

• Grand Duchy of Luxembourg Rue des Tanneurs 7 L-6790 Grevenmacher

Bibliografia

• [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 mconstellations, (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).

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).