The prime-counting function is commonly denoted by $\pi(x)$, and if $b$ is a modulus and $a$ a unit in $\mathbb Z/b$, then $\pi(b,a,x)$ counts the primes in $a \mod b$. Yet suppose one wants to count the primes within a subset $S$ of the natural numbers. What would be the corresponding notation? $\pi(S,x)$? What if $S$ is the union of several residue classes modulo $b$?