Zeta Explained Course

Currently Playing: Zeta Explained #21: The Riemann–von Mangoldt Formula Counts Zeros of the Zeta Function

This is the 21st 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 introduces the Riemann–von Mangoldt Formula (proved in 1905): N(T) ~ T/(2π)log(T/(2π)) - T/(2π) as well as a demonstration showing its accuracy at counting zeros, and then a proof. 00:00 - Intro 01:32 - An online table of the nontrivial zeros of the zeta function 03:28 - Demonstration of the formula for T=10000 06:20 - Demonstration of the formula for T=5000 07:37 - Back to the slides 09:18 - Historical discussion of the proof 09:54 - Proof sketch 11:57 - Proof part 1: Getting most of the formula 34:15 - Proof looks almost done! :) But more than half left :( 35:28 - Proof part 2: Showing N(T+1) - N(T) = O(log T) 55:51 - Proof part 3: Bound on ζ'(s)/ζ(s) 1:11:03 - Final part