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

Tóth László

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 ...

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

Wolf Marek

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 = ...