I want to be able to show that,
\begin{equation} \sum_{p} \log(p) p^{-s} = s \int_{1}^{\infty} \frac{\theta(t)}{t^{s+1}} dt \end{equation} where $\theta(x) =\sum_{p \le x} \log p$. and $\theta(x) = O(x)$.
Using Abel's Summation formula, letting $a(n) = \log(n)$ and $\phi(n) = n^{-s}$,from the definitions here https://en.wikipedia.org/wiki/Abel%27s_summation_formula
I have,
\begin{equation} \sum_{1 \le p \le x} \log p (x^{-s}) - \int_{1}^{x} \sum_{1 \le p \le x} \log u (u^{-s})' du \end{equation}
Is this the right formula to use?