PSNC Repository

Object structure

primes, twins, Skewes number

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.

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