A Simple Proof Of The Prime Number Theorem

TL;DR

This paper demonstrates that basic properties of the zeta function and the Mean Value Theorem can be used to construct simple proofs of the Prime Number Theorem and Dirichlet Theorem, providing more accessible approaches to these fundamental results.

Contribution

It introduces simplified proofs of the Prime Number Theorem and Dirichlet Theorem using elementary tools like the Mean Value Theorem and properties of the zeta function.

Findings

Proofs of the Prime Number Theorem are simplified using elementary methods.

The approach also yields a simple proof of Dirichlet Theorem.

The methods lead to the most straightforward proofs of these asymptotic formulas.

Abstract

It is shown that the Mean Value Theorem for arithmetic functions, and simple properties of the zeta function are sufficient to assemble proofs of the Prime Number Theorem, and Dirichlet Theorem. These are among the simplest proofs of the asymptotic formulas of the corresponding prime counting functions.