Question

Please consider the (presumably infinite) Euler product over the twin primes:

$$ f(z) = \prod_{p\in\mathbb{P}}^{\infty} \Big( 1 - \frac{1}{(p(p+2))^ z} \Big) $$ (in which $p(p+2)$ is a divisor of $4((p-1)!+1) + p$ ).

The Euler Product is a product of a corresponding Dirichlet series. Which one is that?

Thanks in advance,

Max

Edit Update: error fixed.

Answer 1

In this paper: http://arxiv.org/abs/0902.4352, the authors discuss the analytic properties of the related Dirichlet series

$$ D_{2r}(s) = \sum_{n=1}^\infty \frac{\Lambda(n)\Lambda(n+2r)}{n^s}$$

where $\Lambda(\cdot)$ is von Mangoldt's function. See section $2$.