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Volume 17 (1-2) 2011, 87-92

The Skewes Number for Twin Primes: Counting Sign Changes of π2(x) – C2Li2(x)

Wolf Marek

Group of Mathematical Methods in Physics
University of Wrocław
Pl. Maxa Borna 9, PL-50-204 Wrocław, Poland
e-mail: mwolf@ift.uni.wroc.pl

Received:

Received: 05 April 2011; revised: 12 June 2011; accepted: 15 August 2011; published online: 5 October 2011

DOI:   10.12921/cmst.2011.17.01.87-92

OAI:   oai:lib.psnc.pl:742

Abstract:

The results of computer investigation of the sign changes of the difference between the number of twin primes π2 (x) and the Hardy-Littlewood conjecture C2Li2 (x) are reported. It turns out that d2 (x) = π2 (x) − C2Li2 (x) changes the sign at unexpectedly low values of x and for x < 248 = 2.81… × 1014 there are 477118 sign changes of this difference. It is conjectured that the number of sign changes of d2 (x) for x ∈(1, T ) is given by T log(T). The running logarithmic densities of the sets for which d2 (x) > 0 and d2 (x) < 0 are plotted for x up to 248.

Key words:

primes, Skewes number, twins

References:

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