What is the sum of negative integer powers of all prime numbers?

Question

The sum of the reciprocals of all primes, $2^{-1}+ 3^{-1}+5^{-1}+7^{-1}+ \cdots$ is infinite, and the sum $2^{-2}+ 3^{-2}+5^{-2}+7^{-2}+ \cdots$ is $0.4522\cdots$ Moreover, $2^{-3}+3^{-3}+5^{-3}+7^{-3}+ \cdots=0.1747\cdots$ etc. What is known about these numbers?

Answer 1

Coming from my so old cookbook.

Let $$S_n=\sum_{k=1}^\infty \frac 1{p_k^n}=P(n)$$ where $P(n)$ is the prime zeta function. $$\left( \begin{array}{cc} 2 & 0.45224742004106549851 \\ 3 & 0.17476263929944353642 \\ 4 & 0.07699313976424684494 \\ 5 & 0.03575501748392425713 \\ 6 & 0.01707008685063651295 \\ 7 & 0.00828383285613359254 \\ 8 & 0.00406140536651783056 \\ 9 & 0.00200446757496245066 \\ 10 & 0.00099360357443698022 \\ \end{array} \right)$$

Have a look here. If the link is broken, open Wolfram Alpha and enter

DiscretePlot[Log[PrimeZetaP[n]],{n,2,50}]

Have also a look at this beautiful paper from year $1882$ and this one by Euler.