孪生素数的Skewes数：计算$π_2(x)-C_2 \Li_2(x)$的符号变化
The Skewes number for twin primes: counting sign changes of $π_2(x)-C_2 \Li_2(x)$
摘要：
本文报道了孪生素数$π_2(x)$与Hardy-Littlewood猜想$C_2\Li_2(x)$之差的符号变化的计算机研究结果。结果表明，$d_2(x)=π_2(x) - C_2\Li_2(x)$在$x$的出乎意料的低值处改变符号，并且对于$x<2^{48}=2.81\...\times10^{14}$，该差异有477118个符号改变。证明了$x\in (1, T)$的$d_2(x)$的符号变化数由$\sqrt T/\log(T)$给出。绘制了$x$至$2^{48}$的$d_2(x)>0$和$d_2(x)<0$组的运行对数密度。
英文摘要：
The results of the computer investigation of the sign changes of the difference between the number of twin primes $π_2(x)$ and the Hardy--Littlewood conjecture $C_2\Li_2(x)$ are reported. It turns out that $d_2(x)=π_2(x) - C_2\Li_2(x)$ 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 $2^{48}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
