Analog of the Skewes number for twin primes

Details Analog of the Skewes number for twin primes Analog of the Skewes number for twin primes

2007-07-06

arxiv.org/abs/0707.0980v2

Mathematics

Number Theory

Changes: New Figure 1 and a few sentences of justification in favor of the conjecture (5) are made

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 $\pi_2(x) - c_2\Li_2(x)$ 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 $\pi_2(x) - c_2\Li_2(x)$ for $x\in (1, T)$ is given by $\sqrt T/\log(T)$.

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