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

TL;DR

This paper reports on extensive computational analysis of sign changes in the difference between twin prime counts and the Hardy-Littlewood conjecture, revealing unexpectedly early sign changes and proposing a conjecture for their distribution.

Contribution

It provides the first large-scale computational investigation into the sign changes of the twin prime difference and introduces a conjecture relating their count to 24;logarithmic growth.

Findings

Sign changes occur at surprisingly low values of x.

There are 477118 sign changes for x < 2^48.

A conjecture relates the number of sign changes to 24;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 $d_2(x)=\pi_2(x) - C_2\Li_2(x)$ changes the sign at unexpectedly low values of $x$ and for $x<2^{48}=2.81\...\times10^{14}$ there are 477118 sign changes 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 $2^{48}$.