PrimeZetaP function appears to give results for complex
s with real part > 0. Apparently, the analytic continuation is built into the programming. Can anyone explain the mathematical calculation behind this function for $0<\sigma\leq 1?$ The analytic continuation of
Zeta[s] is explained in many texts, but that is not the case for the so-called prime zeta function. I looked at the MathWorld article, which links to a paywalled article, and looked at the online Mathematica guide.
I discussed the prime zeta function at some length in this math.SE answer. In particular, the infinite Möbius inversion
is the actual computational formula used, as recommended in Fröberg's paper. (It is also noted there that numerical evaluation becomes more difficult at values near the imaginary axis, where there is a dense fence of singularities.)