Assuming there exist infinite prime twins does $\prod_i (1+\frac{1}{p_i})$ diverge?

Question

Assume there are an infinite amount of prime twins. Let $p_i$ be the smallest of the $i$ th prime twin. Does that imply that $\prod_i (1+\frac{1}{p_i})$ diverges ?

Answer 1

Viggo Brun showed around $1915$ that the sum of the reciprocals of the prime twins converges. That implies that your product converges.

Answer 2

Based on André Nicolas hint I realized :

$\prod_{i=1}^k (1+\dfrac{1}{p_i}) < (\sum_{i=1}^k (1+\dfrac{1}{p_i}))^2$

And by Brun's theorem it follows the product converges.

Q.E.D.

mick