Convergence of $\sum_{k=1}^\infty p_k^{-z}$, $p$ is prime.

Question

My question is how to show that this $$\sum_{k=1}^\infty p_k^{-z}$$ converges for $\Re(z)>1$. I find it hard to believe since the sum of reciprocals of primes i.e. $z=-1$ diverges, but the key is probably hidden in the fact that we consider only the positive real part of the complex number z.

Answer 1

We know that $\sum_{k=1}^{\infty}|p_k^{-z}|\leq \sum_{k=1}^{\infty} |k^{-z}|$. The right hand side converges for $\mathcal{Re}(z)>1$(this is a well known fact in analysis, and can be seen by applying the integral test on the series). Hence, the left hand side needs to converge too. As $\sum_{k=1}^{\infty}p_k^{-z}\leq \sum_{k=1}^{\infty}|p_k^{-z}|$, we get that $\sum_{k=1}^{\infty}p_k^{-z}$ converges.