I am assuming that the following function, for which I am asking as reference request, should be known in the literature, since Glaisher studied the Prime Zeta Function, and my computation is the first step of Riemann's trick for the deduction of the integral representation of $\zeta(s)\Gamma(s)$, for $\Re s>1$.
Then when I do the substitution $t=p_nu$, denoting $p_n$ as the nth prime number, for $\Re s>1$, in the integral representation formula for the Gamma function you get $$\Gamma(s)P(s)=\int_0^\infty u^{s-1}\sum_{n=1}^\infty e^{-p_n u}du,$$ where $P(s)$ is the Prime Zeta Function.
Question. I am assuming that I can do previous computation, since the integral is absolutely convergent? Can you give a justification to do the change the signs of the series and the integral? After, from a divulgative viewpoint, it is known the function $$\hat{\theta}(u)=\sum_{p\text{ prime}}e^{-p u},$$ or can you say where is defined, or some well known and easy facts about this*? THanks in advance.
$*$ I say if if it possible know if such function satisfies a equation, or if it known how grows, limits or bounds (the only thing from previous link that I can deduce is that $\lim_{s\to1^+}\int_0^\infty u^{s-1}\hat{\theta}(u)du\sim\log \left( \frac{1}{s-1} \right) $, since $\Gamma(1)=1$, but it don't say nothing to me about $\hat{\theta}(u))$. Then on assumption that $\hat{\theta}(u)$ can be defined, I am assuming that it is known from the literature and you can do a short summary about the more important facts: where is defined and some important fact about this special function, or a site where I can see those.