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There is an article on Wikipedia about prime zeta function (PZF):

https://en.m.wikipedia.org/wiki/Prime_zeta_function

In that article , there is table of fairly accurate values of PZF for different $s$ .

We all know that

$\zeta(2n)=π^n\mathbb{Q}$.

So my question is :

Are there conjectured values of PZF such that they are the combination of well known transcendental numbers like $\pi$ and $e$ (like $\zeta(2n)$ above) and are very close to values given in the article ?

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  • $\begingroup$ since these numerical PZF values are on OESIS, I presume if one of these had a closed form expression it would be mentioned there, don't you think so? $\endgroup$
    – Carlo Beenakker
    Apr 28, 2020 at 19:52
  • $\begingroup$ That would be OEIS, e.g., oeis.org/A085548 for digits of prime zeta function at 2. $\endgroup$
    – Gerry Myerson
    Apr 28, 2020 at 23:55

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Hint:As montioned above in the comment by @Carlo Beenakker , the formula is already montioned in the OEIS using Möbius Function which it is $ PZF(p)= \sum_{n=1}^{\infty}\frac{\mu(n)\log \zeta(p^n)}{n}$ its seems that is deduced using Moibus inversion formula with $p$ is a prime number you want to evaluate its zeta prime function

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