Determine whether the following series is convergent or not

Question

Let we have the following series $$\sum_{p\ \text{is prime}}\frac{1}{p}$$

Is there a study on this series whether it is convergent or not , and if it is convergent what its sum is ?

Answer 1

the following idea should allow you to see why $\sum \frac1{p}$ must diverge.

since all the terms are positive and less than 1, the convergence of $\sum \frac1{p}$ implies that of the products $\prod (1-\frac1{p})$ and $\prod \frac1{1-\frac1{p}}$.

however, it is well-known that for $s \gt 1$ $$ \prod \frac1{1-\frac1{p^s}} = \zeta(s)=\sum_{n=1}^{\infty}\frac1{n^s} $$ so the convergence of $\prod \frac1{1-\frac1{p}}$ would imply that $$ \lim_{s \to 1} \zeta(s) = \sum_{n=1}^{\infty} \frac1{n} $$ is finite, which is well-known (and easily proved) not to be the case