Zeta Explained Course

Currently Playing: Zeta Explained #30: The Riemann–Siegel Z Function Under the Riemann Hypothesis

This is the 30th 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. This particular video follows up the previous video where we used the Riemann–Siegel Z function to visualize the Riemann Zeta function on the critical line, and what the Z function should look like if the Riemann Hypothesis is true or false. In particular, we will prove the claim that "if the Z function were to approach zero and turn back without crossing it, that would violate the Riemann Hypothesis." We follow the book "Riemann's Zeta Function" by H. M. Edwards. 00:00 - Intro 02:15 - Z function visualization (Riemann–Siegel Z function) 04:18 - First new graphs (not from last video), of logarithmic derivative of the Z function 07:05 - Proof Sketch 12:29 - Start of technical proof 17:01 - Using the Hadamard product formula for ξ'(s)/ξ(s) 24:20 - Examining the monotonicity of ξ'(s)/ξ(s) 31:15 - Proof end 32:05 - Final visualizations