Prime Number Theorem

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Introduction

The aim of this article is to present the proof of the prime number theorem that says that the number $\pi(n)$ of primes not exceeding $n$ is approximately $\frac n{\ln n}$ or, more precisely, that

$\lim_{n\to\infty}\frac {\pi(n)\ln n}{n}=1.$

Unfortunately, all known proofs of the prime number theorem are quite involved and require knowledge of some college level material.

We shall start with some elementary observations due to Chebyshev.

Chebyshev's estimates

The key observation is that ${2m\choose m}=\frac{(2m)!}{(m!)^2}$ is divisible by all primes $p$ satisfying $m<p\le 2m$ . Indeed, every such prime appears in the numerator $(2m)!$ and does not appear in the denominator $(m!)^2$ . Thus, $\prod_{p\mathrm{\ prime,}\,m<p\le 2m}p\le {2m\choose m}<2^m$ .

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Prime Number Theorem

Introduction

Chebyshev's estimates