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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 $t(n)$ ?

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  • $\begingroup$ Apparantly there is no prime twin or prime gap tag. $\endgroup$
    – mick
    Nov 22, 2015 at 16:21
  • $\begingroup$ Needless to say this question become meaningful for Large $n$ since $t(n) << n$ is required. $\endgroup$
    – mick
    Nov 22, 2015 at 17:11
  • $\begingroup$ 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 ... $\endgroup$
    – mick
    Dec 17, 2019 at 17:50

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