Integral expression for $\log (\zeta (s))$

Question

I read here (and in many other places) that

$$\log (\zeta (s))=s \int_0^{\infty } J(x) x^{-s-1} \, dx$$

where $J(x)=\sum _{n=1}^{\infty } \frac{\pi \left(x^{1/n}\right)}{n}$ is Riemann's prime counting function (link), and $\pi (x)$ is the usual prime counting function. Clearly, the RHS has poles wherever $\zeta (s)=0$.

My question is: what kind of integral is this?

1. Does it in any way relate to the area under the curve $f(x,s)=J(x) x^{-s-1}$? (I doubt it; it seems more likely It's a measure some path in the complex plane.)
2. In what sense can it be said to converge? From rough plotting in Mathematica, it looks to me as though $f(x,s)=J(x) x^{-s-1}$ diverges for values of $s$ less than roughly $-\frac{1}{4}$.
3. Riemann was creating an analytic continuation of the zeta function: so, how does this integral representation cope with complex values of ${x,s}$?

I'm sure these are naive questions, but I am sincerely trying to understand the mechanics of this equation. I'd of course be very grateful for any explanation. In addition, if someone could point me to a source that would allow me to learn and understand this kind of integral, I'd really appreciate it.

Many thanks.

Answer 1

For $\Re(s) \ge 1,s\ne 1$ and not for any other $s$

$$\frac{\log \zeta(s)}{s}=\int_0^\infty J(e^u) e^{-su}du$$ is the Laplace-Fourier transform of $J(e^u) = \sum_{p^k\le e^u} \frac1k$ and $\log \zeta(s)$ is the Laplace-Fourier transform of its distributional derivative $\sum_{p^k} \frac1k \delta(u-k\log p)$.