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The results of the computer investigation of the sign changes of thedifference between the number of twin primes $\pi_2(x)$ and theHardy--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 ofthis 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 logarithmicdensities of the sets for which $d_2(x)>0$ and $d_2(x)<0$ are plotted for $x$up to $2^{48}$.

The Skewes number for twin primes: counting sign changes of $\pi_2(x)-C_2 \Li_2(x)$