孪生素数的Skewes数的模拟
Analog of the Skewes number for twin primes
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。结果是，$π_2(x) - c_2\Li_2(x)$在$x$的出乎意料的低值处改变符号，并且对于$x<2^{42}$，存在超过90000次该差异的符号改变。证明了对于$x\in (1, T)$，$π_2(x) - c_2\Li_2(x)$的符号变化数由$\sqrt T/\log(T)$给出。
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 $π_2(x) - c_2\Li_2(x)$ changes the sign at unexpectedly low values of $x$ and for $x<2^{42}$ there are over 90000 sign changes of this difference. It is conjectured that the number of sign changes of $π_2(x) - c_2\Li_2(x)$ for $x\in (1, T)$ is given by $\sqrt T/\log(T)$.
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