Analog of the Skewes number for twin primes

TL;DR

This paper reports extensive computational evidence on the sign changes of the difference between twin prime counts and the Hardy-Littlewood conjecture, suggesting a new analog of the Skewes number for twin primes.

Contribution

It provides the first large-scale computational investigation into the sign changes of the twin prime counting difference and proposes a conjecture for their distribution.

Findings

Over 90000 sign changes for x<2^42

Sign changes occur at unexpectedly low x values

Conjecture: number of sign changes ~ sqrt(T)/log(T)

Abstract

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 $\sqrt T/\log(T)$.