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.