  Let $\mu(n)$ denote the M\"obius function, define $M(x)= \sum_{n\leq x}^{}\mu
(n)$. The main result of this paper is to prove that \begin{equation*}
\displaystyle\lim_{x \to +\infty}\frac{M(x)}{x}=0 \end{equation*} which is
equivalent to the prime number theorem. We also use Selberg's asymptotic
formula, but the treatments of key parts are different from several classical
proofs.
