Up to now, our examples of arithmetic functions \(f(n)\) have been clearly based on some property of the number \(n\) itself, such as the 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 particular relation to the input — yet in the aggregate bears amazing relations to the input! It is the most mysterious of all these functions.
Definition21.0.1
The prime counting function \(\pi(x)\) is defined, for all positive numbers \(x\) as \[\pi(x)=\#\{p\leq x\mid p\text{ is prime }\}\, .\]