What about $\mathbb{N}\boldsymbol{'}$ for the set of prime numbers in the monoid $\mathbb{N}$? :)
And, of course $M\boldsymbol{'}$ for primes in a monoid $M$. It could even be shorthened to just $\boldsymbol{'}$ when it's understood that the monoid is $\mathbb{N}$.
Let's see:
$$\zeta(s)=\prod_{p\in \mathbb{N}\boldsymbol{'}}\frac{1}{(1-p^{-s})}$$
or even just
$$\zeta(s)=\prod_{\,p \;\,\boldsymbol{'}}\frac{1}{(1-p^{-s})}$$
or
$$\zeta(s)=\prod_{p \;:\,\boldsymbol{'}}\frac{1}{(1-p^{-s})}$$