A math course explaining the Riemann zeta function
Curated by: Zeta Explained (121 videos)
This is the 36th 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 analyzes the growth rate of the zeta function for a fixed real part at least one and taking the imaginary part to infinity. In particular, we show that zeta(1+it) = Omega(log log t) using the Dirichlet Approximation Theorem and the Phragmén–Lindelöf Principle. 00:00 - Intro and visualization 01:50 - Omega definition and disambiguation 06:41 - Zeta function 08:12 - Theorem 1: For a fixed sigma greater than 1, zeta(sigma+it) gets arbitrarily close to zeta(sigma) for large t. 22:49 - Corollary 1: zeta is unbounded in Re(s) greater than 1 and Im(s) greater than delta 25:14 - Corollary 2: zeta(sigma+it) greater than A*(log log t) 29:16 - Theorem 2: zeta(1+it) = Omega(log log t) 38:52 - Using an approximate functional equation to justify visualizing the function with a partial sum