Do all prime numbers and only prime numbers have this property:
$p$ is a factor of the simplified denominator of $ζ(-p + 2)$?
I was looking at the values of the Reimann-zeta function and noticed this property. I'm not familiar with this function and don't have a background in math, so I was wondering if someone would be able to chime in :).
Note that $ζ(-n + 2)=\frac{B_{n-1}}{n-1}$ for all $n \ge 2$ (so in particular $\zeta(0)=-1/2$). The Theorem of Clausen and Von Stadt tells us precisely how the denominator of $B_n$ looks for $n$ even namely
$B_n=-\sum_{p-1|n}\frac{1}{p}+C_n$ where the sum is taken on all primes st $p-1|n$ and $n$ even and at least $2$ while $C_n$ is some integer. Hence for $p$ odd prime we clearly have that $p$ appears in the denominator of $B_{p-1}$ hence of $\zeta(2-p)$ while for $p=2$ the result also holds as noted