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

Tóth L.

Computational Methods in Science and Technology

2019

Vol. 25, No. 3

143--148

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.

The Skewes Number for Twin Primes: Counting Sign Changes of pi 2(x) - C2Li2(x)

Wolf M.

Computational Methods in Science and Technology

2011

Vol. 17, No. 1-2

87-92

The results of computer investigation of the sign changes of the difference between the number of twin primes pi2 (x) and the Hardy-Littlewood conjecture C2Li2 (x) are reported. It turns out that d2 (x) = pi2 (x) - C2Li2 (x) changes the sign at unexpectedly low values of x and for x less than 248 = 2.81... x 1014 there are 477118 sign changes of this difference. It is conjectured that the number of sign changes of d2 (x) for x element of (1, T ) is given by T log(T). The running logarithmic densities of the sets for which d2 (x) greather than 0 and d2 (x) less than 0 are plotted for x up to 2 48.