prove $\sum_p \frac{\log p}{p^s} = \int_{1}^{\infty}\frac{d\theta(x)}{x^s}$

Question

I am currently studying Zagier's paper, Newman's Short Proof of the Prime Number Theorem, found here https://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2975232/fulltext.pdf . In proof (V) of his paper, he asserts that $$\sum_p \frac{\log p}{p^s} = \int_{1}^{\infty}\frac{d\theta(x)}{x^s},$$ where $\theta(x) = \sum_{p \leq x} \log p$ for prime $p$. Being taken back by this a bit, I did some research, which has lead me to believe that this has to do with Stieltjes integral's, which I have not heard of until now. I have not quite been able to figure out explicitly why this equation is true, despite my trivial understanding of Stieltjes integral's. I'm wondering if anyone can point me in the right direction.

Answer 1

Because $\theta(x)$ only increases when $x$ is prime and there, the function increases by $\log p$.