Zeta Explained Course

Currently Playing: Zeta Explained #48: Proof of the Prime Number Theorem

This is the 48th video in a series explaining the Riemann zeta function. The idea of the series is to start with basics and eventually work our way to advanced topics. We follow the book "The Riemann Zeta-Function: Theory and Applications" by Aleksandar Ivić. This particular video covers the proof of the Prime Number Theorem using the truncated explicit formula for Chebyshev's ψ(x) function which counts up by log p on prime powers. We then convert the results about ψ(x) ~ x into π(x) ~ x/(log x). Then we go over the error term and what this implies about the distribution of prime numbers, i.e. that there can't be too many prime numbers bunched up together, and there can't be too long of a gap without prime numbers. We also go over how improving the zero-free region of ζ(s) improves the error term in the PNT. 00:00 - Intro 06:00 - Zero-free regions (de la Vallée Poussin) 08:40 - A lemma on 1/|ρ| 11:51 - Converting the zero-free region into an error bound for ψ(x) ~ x 20:45 - Converting ψ(x) ~ x into π(x) ~ x/(log x) 30:42 - Graphs of prime counting functions π(x), θ(x), ψ(x) 33:19 - The error term in the PNT regulates the distribution of primes 35:08 - Visualizing the ψ(x) error bound 43:32 - A theorem about the PNT error bound being equivalent to uniform zero-free regions of ζ(s) 48:13 - ψ(x) = x + O(sqrt(x) log^2 x) is equivalent to the Riemann Hypothesis 51:14 - Visualizing the π(x) error bound with RH 52:25 - Modern zero-free regions (Vinogradov-Korobov)