Prime Number Theorem

Let \(\pi(x)\) be the number of primes less than or equal to \(x>1\). The prime number theorem asserts that
\[ \pi(x) \bigg/ \frac{x}{\log{x}} \rightarrow 1 \quad (x \rightarrow \infty). \]
Historically, Jacques Hadamard and Charles Jean de la Vallée-Poussin independently proved this theorem in 1896. They made full use of the Riemann zeta function \[ \zeta(s) := \sum_{n=1}^\infty \frac{1}{n^s}, \quad s\in\mathbb{C}, \quad \operatorname{Re}(s)>1 \] and complex analysis for this. After that , Atle Selberg and Paul Erdös independently proved this theorem only by using calculus of one-variable in 1949.

Prime Number Theorem pnt.jl