I searched for primes of the form of $\lfloor \zeta(-n) \rfloor$, where $n \in \Bbb{N}$, for a range of $n \le 10^4$ on PARI/GP and found $\lfloor \zeta(-n) \rfloor$ is only prime for $n=23$.

My PARI code:

for(n=1, 10^4, if(ispseudoprime(floor(-bernfrac(2*n)/(2*n)))==1, print([2*n-1, floor(-bernfrac(2*n)/(2*n))])); print(2*n-1)) 

Note that $\zeta(-n)$ for odd $n$ (for even $n$, $\zeta(-n)=0$) can be also expressed as:

$$\zeta(-(2n-1))=-\frac{B_{2n}}{2n}$$ where $B_{2n}$ is the $2n$th Bernoulli number and written as brenfrac(2*n) in PARI/GP.


$(1)$ Is $\lfloor \zeta(-n) \rfloor$, where $n \in \Bbb{N}$, only prime for $n=23$?

$(2)$ Are there finite primes of the form of $\lfloor \zeta(-n) \rfloor$?

I would appreciate any counterexamples(can be a probable prime)/proofs/papers.

Extra: I also searched for primes of the form $\lceil\zeta(-n)\rceil$ and $\lfloor\zeta(-n)\rceil$. For $n \le 10^4$:

$(1)$ $\lceil\zeta(-n)\rceil$ seems to be only prime for $n=691$.

$(2)$ There seems to be no prime of the form $\lfloor\zeta(-n)\rceil$.

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    $\begingroup$ Update: nothing below $1.22\times 10^4$ for the "ceiling" and "floor" versions (other than the $2$ numbers in the OP's post). Had to stop the computations due to unforeseen circumstances. Might continue tomorrow. $\endgroup$ – Klangen Jun 10 at 16:36
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    $\begingroup$ @GerryMyerson The floor function makes it an integer. $\endgroup$ – Mathphile Jun 11 at 11:08
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    $\begingroup$ Do you have any good reason to look at $\lfloor B_{2n}/2n\rfloor$ ? The Bernouilli numbers are complicated, there are theorems proving modulo and growth properties, most of them are very deep (they fit in the same theory as Fermat last theorem..) $\endgroup$ – reuns Jun 11 at 20:46
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    $\begingroup$ Very often many properties of primes occur with a $\log\log x$ density hence they give the impression that they are extremely rare or may have finitely many solutions, while in reality there are infinitely many solutions which are appear to be rate because of the $\log\log x$ frequency of occurrence. This may or may not be the case here but we must exercise caution in looking at something about primes. E.g. The smallest prime $p_n \equiv 330 \mod n $ is $p_n = p_{1208198749}$ $\endgroup$ – Nilotpal Kanti Sinha Jun 12 at 4:42
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    $\begingroup$ Update: $\lfloor \zeta(-n) \rfloor$ has no other prime upto $n \le 2 \times 10^4$ $\endgroup$ – Mathphile Jun 13 at 13:09

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