A math course explaining the Riemann zeta function
Curated by: Zeta Explained (121 videos)
This is the 28th 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. The viewer is expected to understand calculus and complex numbers, whereas I will try to explain concepts from complex analysis as needed. We will follow the books "The Riemann Zeta Function: Theory and Applications" by Aleksandar Ivić and "Riemann's Zeta Function" by H. M. Edwards. This particular video follows up Ep. 27 by walking through a proof of the Prime Number Theorem by doing complex analysis on the Riemann zeta function. In particular, the final result uses the nontrivial zeros of the Riemann zeta function. 00:00 - Intro 01:16 - Intro to Chebyshev's functions θ(x) and ψ(x) 02:36 - Graphs of Gauss and Chebyshev functions 04:39 - Two formulas involving the von Mangoldt function 06:55 - Perron's inversion formula and Dirichlet series 11:36 - Writing ψ(x) in terms of ζ'(s)/ζ(s) 14:49 - Recap of Riemann zeta function 16:00 - A contour integral 17:05 - Computing the residues 23:09 - Recap of the residues 28:04 - Final term remaining in proof of PNT, involves nontrivial zeros 32:35 - Apologies, de la Vallée Poussin was Belgian*, not French 34:51 - Riemann hypothesis implies a stronger bound on PNT 35:51 - Recap of last two episodes 37:27 - Zero-free regions