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.
We can use the following
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 $$\sum_{n\geq1}a_{n}=\sum_{p,p+2\in\mathfrak{P}}\left(\frac{1}{p}+\frac{1}{p+2}\right)=B\approx1.90216. $$