The Prime Counting Function

Definition: The Prime Counting Function is the function $\pi : \mathbb{R} \to \mathbb{N}$ defined for all $x \in \mathbb{R}$ by $\displaystyle{\pi (x) = \sum_{p \leq x} 1}$.

The prime counting function adds $1$ for every prime less than or equal to $x$.

Note that the prime counting function can take on non-integer values. For example, $\pi (1.4) = 0$. The interesting values of $\pi(x)$ are of course $x \in \mathbb{N}$. Some values of $\pi(x)$ are given in the table below.

$x$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$\pi (x)$	$0$	$1$	$2$	$2$	$3$	$3$	$4$	$4$	$4$	$4$

Note that the prime counting function is not multiplicative. To see this, note that $(2, 3) =1$, but $\pi (2) \pi (3) = (1)(2) = 2$ while $\pi (6) = 3$.