\begin{definition}[Definition:Prime-Counting Function]
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:
:$\ds \forall x \in \R: \map \pi x = \sum_{\substack {p \mathop \in \mathbb P \\ p \mathop \le x} } 1$
where $\mathbb P$ denotes the set of prime numbers.
\end{definition}