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Preamble: I asked this question on Math.SE with a bounty of 100, but received no replies. The bounty has now ended, and it was suggested I post this here instead.


Since Apéry we know that $\zeta(3)$, where $\zeta$ denotes the Riemann zeta function, is irrational. It is also well known that infinitely many values of the Riemann zeta function at odd positive integers are irrational. Moreover, various results by Zudilin have shown that certain subsets of zeta values at odd positive integers are irrational; for instance, at least one of $ζ(5), ζ(7), ζ(9)$, or $ζ(11)$ is irrational.

Are there any similar results for $P(n)$, where $P$ is the prime zeta function, i.e.,

$$ {\displaystyle P(n)=\sum _{p\,\in \mathrm {\,primes} }{\frac {1}{p^{n}}}={\frac {1}{2^{n}}}+{\frac {1}{3^{n}}}+{\frac {1}{5^{n}}}+{\frac {1}{7^{n}}}+{\frac {1}{11^{n}}}+\cdots ?} $$

A quick search on Wolfram Alpha reveals the following:

I was not able to find any papers or articles related to the irrationality of values of $P$ at positive integers. Have these been studied in a (more or less) serious manner, analogously to $\zeta$? What are the current results?


EDIT: In the comment section several links to Math.SE have been posted. These do not answer my question, since I am looking for recent research on the subject, not answers of the type "no" or "yes".

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    $\begingroup$ The number of proofs that we have of showing some numbers are irrational are very limited. We either show a number $\alpha$ is irrational because it is algebraic of degree greater than one (by exhibiting an irreducible polynomial $f$ of degree greater than one $f(\alpha) = 0$), or we find a sequence of rational numbers that converge to $\alpha$ way too fast (basically the idea behind the transcendence proofs of $\pi$ and $e$, and the irrationality of $\zeta(3)$). The numbers you wrote down don’t seem to be attackable by either approach, so showing that they are irrational is very hard. $\endgroup$ – Stanley Yao Xiao Oct 18 '18 at 10:04
  • $\begingroup$ I don't know what you mean by Math.SO, but whatever it may be, you should include a link here to your post there (and a link there to your post here). Anyway, I don't think anything is known about irrationality of $P(n)$. No one knows a way to get a handle on it, and no one cares too much, since $P(n)$ doesn't seem to be related to other stuff the way $\zeta(s)$ is. $\endgroup$ – Gerry Myerson Oct 18 '18 at 12:10
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    $\begingroup$ @GerryMyerson Thank you for your response. The link is already in the preamble (just click on "Math.SE") $\endgroup$ – Klangen Oct 18 '18 at 12:34
  • $\begingroup$ @GerryMyerson Concerning your last sentence, since ${\displaystyle \log \zeta (s)=\sum _{n>0}{\frac {P(ns)}{n}}}$, I think it is interesting enough. $\endgroup$ – Klangen Oct 18 '18 at 12:36
  • $\begingroup$ Possible duplicate of Convergence of the series $\sum_p p^{-s}$ ($p$ prime and $s>1$) $\endgroup$ – Gerry Myerson Oct 18 '18 at 12:49

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