The sum asymptotics when $x\to \infty$, obviously, must be a function of two variables, $x$ and $s$ :
$$\sum\limits_{p\leqslant x}^{}\frac{1}{p^s}=F(x,s)$$
However, as for the sum of inverse powers of prime numbers, this function is remarkable of the presence of the self-similiarity, that is, if we denote $ \pi (x) $ - prime counting function, then for the asymptotic behavior of the sum , apparently, will have:
$$F(x,s)=\sum\limits_{p\leqslant x}^{}\frac{1}{p^s}=\pi(x^{1/(1-s)})$$
This formula was confirmed experimentally in the range of $0\leqslant s <1$ and it follows from a known approximation $\pi(x)$ by logarithmic integral function. Obviously, this property is about to be discussed somewhere, I would like to read about this in more detail.
Yes, I made a mistake, not $\pi(x^{1/(1-s)})$ but $\pi(x^{(1-s)})$: $$F(x,s)=\sum\limits_{p\leqslant x}^{}\frac{1}{p^s}=\pi(x^{(1-s)})$$