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.
Definition21.0.1.
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*}
Summary:The Prime Counting Function
Here, we harness the power of the Legendre symbol to find a deep correlation between solutions of two seemingly unrelated congruences – a correlation that enables us to tell very quickly whether any quadratic congruence has a solution!
Section 21.1 introduces the prime counting function \(\pi(x)\text{.}\)
Section 21.2 gives some history and cool graphics to help suggest there is some regularity in this behavior.
