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Consider the following function :

$$F(s)= \sum_{\text{$p,\ p+2$ are primes}} \left({\frac{1}{p^s}}+{\frac{1}{(p+2)^s}}\right).$$

Brun's theorem tells us that $F(1)$ is finite.

We are looking for the infiniteness of $F(0)$. Indeed, this is the Twin Prime Conjecture, supported by extensive numerical evidence.

Is there a numerical estimate for a value of $s\in(0,1)$ such that $F(s)$ seems to diverge for that value (and all values less than that of course)?

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It is conjectured that the number of twin primes between $x$ and $2x$ has order of magnitude $x/\log^2 x$. If this is true, then $F(s)$ diverges for any $s<1$. See also this related MO post.

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  • $\begingroup$ thank you for the answer sir , but can I ask for reference for table of values for any value of $s$ in given domain ? $\endgroup$ – user145059 Feb 20 '20 at 9:21
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    $\begingroup$ @Alexandersupertramp: Not sure what you mean by "table of values". You mentioned "extensive numerical evidence" for the Twin Prime Conjecture. This also concerns the quantitative version (first sentence in my response). The quantitative version implies that $F(s)$ diverges for $s<1$. When I say "implies" I mean it in a strict mathematical sense. Any numerical estimate for the number of twin primes up to $x$ in dyadic ranges gives a bound for $F(s)$. So it is fair to say that we have "extensive numerical evidence" for the divergence of $F(s)$ for every $s<1$. $\endgroup$ – GH from MO Feb 20 '20 at 9:37

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