Divergence of the series of the inverses of the prime numbers

Question

Can you help me prove that the $\sum \limits _{p \text{ prime}} \dfrac 1 p$ is divergent?

Answer 1

Here is a sketch of a proof. Of course, the hard part lies in the results that are used for this sketch.

A consequence of the Prime Number theorem is the $n$th prime number function $p_n$ is asymptotically equivalent to $\;n\log n$ (actually, it is equivalent to the Prime Number theorem).

Hence $\dfrac1{p_n}\sim \dfrac1{n\log n}$, so that the series $\displaystyle\sum_n\frac1{p_n}\enspace\text{and}\enspace\sum_n\frac1{n\log n}$ both converge or both diverge. However, the latter is a Bertrand's series which diverge.