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arXiv · 2302.12218

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

Junda Pan
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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.

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Junda Pan. 2023-02-21. An Elementary Proof of the Prime Number Theorem based on M\"obius Function. https://arxiv.org/abs/2302.12218

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