# An Elementary Proof of the Prime Number Theorem based on M\"obius Function

## TL;DR

This paper provides an elementary proof of the prime number theorem by demonstrating that the M"obius function's summatory function divided by x tends to zero, offering a different approach from classical proofs.

## Contribution

It presents a novel elementary proof of the prime number theorem using the M"obius function and Selberg's asymptotic formula, with unique treatment of key components.

## Key findings

- Proves that M(x)/x approaches zero as x tends to infinity.
- Establishes the prime number theorem through an elementary approach.
- Uses Selberg's asymptotic formula with a different methodology.

## Abstract

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.