The Prime Number Theorem and Riemann's Zeta Function

Abstract

We give a proof of the Prime Number Theorem using the Riemann zeta function. After establishing the analytical tools required, we approach the main result. We derive an asymptotic formula with error terms for the growth of the prime counting function by de la Vallée-Poussin's zero free region of the zeta function and the explicit von Mangoldt formula. We also add a brief presentation of historical development in regards to the theory of prime numbers.