## 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