# Notation for the prime-counting functions for subsets of the natural numbers

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 $$b$$?

• You can use any notation you like, provided you are careful to explain it. – Gerry Myerson Jan 12 at 23:42