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The 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.

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I discussed the prime zeta function at some length in this math.SE answer. In particular, the infinite Möbius inversion

$$P(s)=\sum_{k=1}^\infty \frac{\mu(k)}{k}\log\zeta(ks)$$

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.)

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  • $\begingroup$ Will check the MSE answer. It's valid on (0,1] then I take it? $\endgroup$
    – daniel
    Nov 19, 2015 at 16:25
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    $\begingroup$ Yes, except at singular points, of course. $\endgroup$ Nov 19, 2015 at 16:26

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