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 < 2 48 = 2.81... × 10 14 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 2 48.
|The Skewes Number for Twin Primes: Counting Sign Changes of π2(x) – C2Li2(x)||2014-08-19|