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

$$\zeta(s)=\prod_{p \;:\,\boldsymbol{'}}\frac{1}{(1-p^{-s})}$$