Dirichlet series for prime sequence

Question

With $p_n = n^{th}$ prime and $f(s):=\sum_{n=1}^\infty 1/p_n^s$ when the series converges. What is the status of the following questions: What is the abscissa of convergence of $f$? What are the values of $f$ at integers > 1?

Answer 1

This is the prime zeta function, see e.g. http://en.wikipedia.org/wiki/Prime_zeta_function. The series converges for $\Re(s) > 1 $. On the Wiki page you can find the some function values for integer arguments.

Answer 2

There is indeed very limited literature on this. And the reason is that such is not been achieved yet. The key classical article on the prime zeta function is that of Fröberg. Then what could intuitively help you to build a broader understanding of the prime zeta function, also in relation to the Riemann zeta function, is a smart lecture over here. What indeed will help you rather understand the "why not" can be intuitively gathered from the relation between the prime zeta function, and "the Riemann zeta function, its derivative and the Möbius function".

In advance to the question "What is the abscissa of convergence?" one must find to a reasonable mathematical foundation, at least a rationale for a conjecture why and when "such abscissa shall be existent"? Then your second question of closed values at integers could follow.

In particular and for instance take into account that, (1) at all squarefree positive integers $\alpha$ we obtain singualrities in the form of bifurcation when either $s=1/\alpha$ or $s=\rho/\alpha$; while $\rho$ a non-trivial zero of the Riemann zeta function.

and (2) that we can gain: $$\sum_{n=1}^\infty \frac{P(ns)}{n}= \log \zeta(s)$$

while $P$ the prime zeta function.

General reference to wikki (and more extended) holds as requested by commentator of this answer.