# Does data suggest $| \pi_2 (n) - 2\Pi \int_2^n \frac{dx}{\ln(x)^2} | < \ln(n+2)^2 \sqrt (n+2)$?

Let $\Pi$ be the twin prime constant and $\pi_2(n)$ the twin prime counting function.

Define

$$t(n) = \left| \pi_2(n) - 2 \Pi \int_2^n \frac{dx}{\ln(x)^2} \right|$$

Is it consistent with current data that for $n > 100$ we have :

$$t(n) < \ln(n+2)^2 \sqrt {(n+2)}$$

??

Is there a stronger plausible boundary for $t(n)$ ?

• Apparantly there is no prime twin or prime gap tag. – mick Nov 22 '15 at 16:21
• Needless to say this question become meaningful for Large $n$ since $t(n) << n$ is required. – mick Nov 22 '15 at 17:11
• I heard the largest prime twin gap is conjectured and tested to be O( ln(n)^3 ). I would have guessed O( ln(n)^4 ) ; the square of the conjectured largest prime gap O(ln(n)^2). Not sure how and if that relates ... – mick Dec 17 '19 at 17:50