Definition:Prime-Counting Function

Definition
The prime-counting function is the function $\pi: \R \to \Z$ which counts the number of primes less than or equal to some real number.

That is:
 * $\displaystyle \forall x \in \R: \map \pi x = \sum_{\substack {p \mathop \in \mathbb P \\ p \mathop \le x} } 1$

where $\mathbb P$ denotes the set of prime numbers.

Also defined as
Some sources give both the domain and codomain as $\N$, thus:
 * $\pi: \N \to \N$

Also known as
Some sources merely call this the $\pi$ function.

Also see

 * Prime Number Theorem