As in other answers and comments: context usually suffices to explain that $p$ is a prime, whether in the rational integers or whatever. That is, when possible, no notation at all is clearer (and less bulky and visually noisy) than any possible notation.
Even in situations where clarification is essential, in-lined expressions can be almost entirely prose, rather than symbolic, and displayed expressions can have a small verbal comment, as in $$\zeta(s)\;=\;\prod_p \frac{1}{1-p^{-s}}\hskip30pt\hbox{(product over primes p)}$$