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Martin Sleziak
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Klangen
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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".

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?

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".

Notice added Authoritative reference needed by Klangen
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Klangen
Klangen
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Preamble: I asked this question on Math.SOSE 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?

Preamble: I asked this question on Math.SO 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?

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?

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Klangen
Klangen
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