For $n\geq 1$ let the nth twin prime pair $$(p_n,p_n+2).$$

This sequence start as $(3,5),(5,7)$, the next $(11,13)\ldots$. I have two short questions about twin primes and infinite product defined from these. Can you clarify my doubts? Thanks in advance.

Question. For $a_n=\frac{1}{p_n}+\frac{1}{p_n+2}$ can be justified the convergence of $$\prod_{n\geq 1}(1-a_n)=\prod_{n\geq 1}\frac{p_n^2-2}{p_n(p_n+2)}?$$

My attempt: Since $1>a_n\geq 0$ and $\sum_{n\geq 1}a_n$ converges by Brun's theorem (see as quick reference this site or for example Wikipedia) then previous product is convergent to a constant c. Is it a formalized proof?

Question. Is is possible to define, at least for $\Re s>1$ (the abscissa of absolute convergence) $$\tau(s)=\prod_{n\geq 1}(1-p_n^{-s})^{-1}?$$

My attempt: Is right say this?: Well as the (classical) Euler product $$\prod_{\text{p:prime}}(1-p^{-s})^{-1}$$ is defined as convergent for this abscissa of convergence $\Re s>1$ and the support for previous case, the case of twin primes is a subset of the support in Euler product, then too there is convergence at least for this abscissa.

Example. For example if two previous products (corresponding previous questions) can be defined, then we can write for example $$\tau(2)=c\cdot\prod_{n\geq 1}\frac{p_n^3(p_n+2)}{(p_n^2-1)(p_n^2-2)},$$

where we are assuming the definition of the pair $(p_n,p_n+2)$ as before, and $c$ is the previous cited constant after first question.

  • $\begingroup$ Note, that it is still unknown, whether infinite many twin-primes exist. But as I understand, we assume that there are infinite many twin-primes. $\endgroup$ – Peter Dec 23 '15 at 11:19
  • $\begingroup$ @Peter At no point we need to assume this. Also note that if there are finitely many twin primes, then the product converges as it's finite. $\endgroup$ – Wojowu Dec 23 '15 at 11:23
  • $\begingroup$ Thanks @Peter, I believe that Bruns theorem don't assume that there are infinitely many primes. To have sense (I believe that in other case we have only finite products) I belive that we should assume that there are infinitely many twin primes, but all discussion are welcome. $\endgroup$ – user243301 Dec 23 '15 at 11:24
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    $\begingroup$ The infinite-product-tag indicates that the OP assumes that there infinite many. Brun's theorem is, of course, also true, if the number of twin-primes is finite. I just wanted to point out that the product in Brun's theorem could be finite. $\endgroup$ – Peter Dec 23 '15 at 11:25
  • $\begingroup$ If in the future someone want to contribute with comments to second question, if it is has mathematical sense I will appreciate this, thanks. My goal sponsor previous question was understand a few more about infinite products. $\endgroup$ – user243301 Jan 1 '16 at 18:14

We can use the following

Theorem. $\prod_{n\geq1}\left(1+b_{n}\right) $ converge absolutely if and only if $\sum_{n\geq1}\left|b_{n}\right|<\infty $.

Then consider $b_{n}=-a_{n} $. So we have to check if $$\sum_{n\geq1}\left|b_{n}\right|=\sum_{n\geq1}\left|a_{n}\right|=\sum_{n\geq1}a_{n}<\infty $$ and this follows by the Brun's theorem, since $$\sum_{n\geq1}a_{n}=\sum_{p,p+2\in\mathfrak{P}}\left(\frac{1}{p}+\frac{1}{p+2}\right)=B\approx1.90216. $$

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  • $\begingroup$ Very thanks much @MarcoCantarini $\endgroup$ – user243301 Dec 30 '15 at 10:53

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