Let the function q(z) of one complex variable z be the sum over all primes p of (1/p^z). I was wondering about the complex zeros of q(z) [hoping that this problem might be much easier than the same problem for the sum over all integers!] and I realized that the first step would be to find an analytic continuation for q(z). In trying to develop that, I realized that I can't even easily prove that q(z) diverges for all Re(z) <= 1 (although it is easy to prove that q(1) diverges).


a) Is there a nice proof that q(z) diverges whenever Re(z) = 1?

b) Can somebody find an analytic continuation for q(z) valid in regions of the complex plane with Re(z) <= 1?

And the really tough question:

c) Is there some unifying feature of the zeros of p(z) (for example, for zeta(z) the zeros probably all lie on one straight line? If necessary, you can assume the Reimann hypothesis for this question.

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    $\begingroup$ (b) Can be done by using (infinite) Mobius inversion and zeta's continuation (look up prime zeta function on Wikipedia). It cannot be extended left of Re=0 however. For (c), I imagine the zeros are quite chaotic and unruly and theoretically unremarkable. I think (a) is a good question though. IIRC, there exist values for which the usual p-series (for zeta) do actually converge, so it's natural to ask that about the prime zeta series. $\endgroup$ – blue May 16 '14 at 0:06
  • $\begingroup$ On the Mathworld page about prime zeta function, there is a mistake! (Unusual for Mathworld) The mistake is that the plot of P(1+it) is the wrong plot, it is a duplicate of P(1/2 + it). $\endgroup$ – Mark Fischler May 19 '14 at 19:03

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