NTIC The Prime Counting Function

Chapter 21 The Prime Counting Function

Up to now, our examples of arithmetic functions

$f(n)$ have been clearly based on some property of the number

$n$ itself, such as its divisors, the numbers coprime to it, and so forth.

However, there is one function of prime importance which, as far as we yet know, bears no particularly obvious relation to the input – yet in the aggregate bears amazing relations to the input! It is the most mysterious one of all.

🔗

Definition 21.0.1.

The prime counting function $\pi(x)$ is defined, for all positive numbers $x\text{,}$ as the number of primes less than or equal to $x\text{,}$ denoted

$\begin{equation*} \pi(x)=\#\{p\leq x\mid p\text{ is prime }\}\text{.} \end{equation*}$