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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 $\pi(x^{(1-s)})$: $$F(x,s)=\sum\limits_{p\leqslant x}^{}\frac{1}{p^s}=\pi(x^{(1-s)})$$

Aleksey Druggist
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    You may take a look here http://math.stackexchange.com/questions/49383/how-does-sum-px-p-s-grow-asymptotically-for-textres-1/49434#49434 – Marco Cantarini Oct 18 '16 at 06:43
  • Marco Cantarini, thanks for the link, but I also would like to see some speculations about the consequences of the self-similarity of this function in broader sense (for example, self-similarity concept is widely used in phase transition theory in physics) – Aleksey Druggist Oct 18 '16 at 10:23

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