Zeta Explained Course

Currently Playing: Zeta Explained #13: Proof that the Zeta Function Has No Zeros on the Line Re(s) = 1

This is the 13th 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 book "The Riemann Zeta Function: Theory and Applications" by Aleksandar Ivić. This particular video covers the argument principle in complex analysis and proves that the zeta function has no zeros on the line with real part 1. 00:00 - Intro 00:45 - Crash course on complex analysis: The Argument Principle and the Laurent series 02:47 - Visualization of the Argument Principle 06:20 - Proof Sketch 07:14 - Start of proof by contradiction 14:24 - Recap of proof so far 15:37 - Proof that a sum is greater than 0 17:13 - Proof that the same sum is at most 0 22:53 - Contradiction