An Elementary Proof of the Prime Number Theorem based on Möbius Function

Let $\mu (n)$ denote the Möbius function, define $M(x)={\sum }_{n\le x}^{}\mu (n)$. 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.