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.