Zeta Explained #26: The Phragmén–Lindelöf Principle and the Order of Zeta in the Critical Strip скачать в хорошем качестве

Zeta Explained #26: The Phragmén–Lindelöf Principle and the Order of Zeta in the Critical Strip

This is the 26th 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 the Riemann-von Mangoldt equation estimating the number of zeros in the critical strip between 0 and T as T/(2π)log(T/(2π)) - T/(2π). 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 Alexandar Ivić. This particular video introduces the Maximum Modulus Principle, and by extension the Phragmén–Lindelöf Principle. We then use the Phragmén–Lindelöf Principle to analyze the growth rate of the zeta function inside the critical strip and get Lindelöf's bound of mu(1/2) is at most 1/4. 00:00 - Intro 00:11 - The Maximum Modulus Principle 07:26 - Intro to the Phragmén–Lindelöf Principle 11:13 - Counterexample to Phragmén–Lindelöf Principle without restriction on function 17:38 - Back to the Riemann zeta function 27:17 - The mu function and relating to the Lindelöf Hypothesis