The series diverges. As Euler showed.

Define to be the primes . For a given (sufficiently large) let be the primes .

Define .

Notice that .

Therefore, we see that .

By the fundamental theorem of arithmetic any number be be written as where .

It follows that,

.

Since the harmonic series diverges it means .

However,

Thus,

Notice that,

We have shown,

Remember that .

What this means is thatifthen would be bounded.

This is impossible, it is not bounded.

Thus, the sum of prime reciprocals must diverge.