Does $\prod_{n=1}^\infty\left(1+\frac{1}{p_n}\right)$ diverge?

Question

$p_n$ is the $n^\text{th}$ prime number.

This looks like Euler's product for zeta function, but I don't know if they are related.

Answer 1

Yes it does diverge.

$$\prod_{n=1}^\infty\left(1+\frac{1}{p_n}\right) \ge 1+\sum_{n=1}^{\infty} \frac{1}{p_n}$$

the RHS of which is well-known to diverge.

In fact, $\sum_{n=1}^{N} \frac{1}{p_n} = \Omega(\log \log N)$.