The Skewes number for twin primes: counting sign changes of $π_2(x)-C_2 \Li_2(x)$

Details The Skewes number for twin primes: counting sign changes of $π_2(x)-C_2 \Li_2(x)$ The Skewes number for twin primes: counting sign changes of $π_2(x)-C_2 \Li_2(x)$

2011-07-14

arxiv.org/abs/1107.2809v1

Computational Methods in Science and Technology vol.17 (2011) pp. 87-92

Mathematics

Number Theory

16 pages, 4 figures

Scientific paper

The results of the computer investigation of the sign changes of the difference between the number of twin primes $\pi_2(x)$ and the Hardy--Littlewood conjecture $C_2\Li_2(x)$ are reported. It turns out that $d_2(x)=\pi_2(x) - C_2\Li_2(x)$ changes the sign at unexpectedly low values of $x$ and for $x<2^{48}=2.81\...\times10^{14}$ there are 477118 sign changes of this difference. It is conjectured that the number of sign changes of $d_2(x)$ for $x\in (1, T)$ is given by $\sqrt T/\log(T)$. The running logarithmic densities of the sets for which $d_2(x)>0$ and $d_2(x)<0$ are plotted for $x$ up to $2^{48}$.

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