Zeta Explained Course

Currently Playing: Zeta Explained #37: Zeta Can Get Close to Zero in Re(s) ≥ 1

This is the 37th 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 "The Theory of the Riemann Zeta Function" by E.C. Titchmarsh. This particular video analyzes how close the Riemann zeta function can get to 0 even in regions where it was proven it cannot be equal to 0. In particular, we show that 1/ζ is unbounded in the region Re(s) ≥ 1, using Kronecker's Approximation Theorem and the Phragmén–Lindelöf Principle. 00:00 - Theorem 1 (unbounded when Re(s)＞1) and proof 08:28 - Using Kronecker's Approximation Theorem to get cosines simultaneously negative 16:10 - Proof summary 17:27 - Theorem 2 (unbounded when Re(s) = 1) and proof using the Phragmén–Lindelöf Principle 22:47 - Issue with finding any specific bounds 25:11 - Visualizations