Zeta Explained #25: The Lindelöf Hypothesis (and the Riemann Hypothesis) скачать в хорошем качестве

Zeta Explained #25: The Lindelöf Hypothesis (and the Riemann Hypothesis)

This is the 25th 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 growth rate of the Riemann zeta function, what the Lindelöf Hypothesis is, and its relation to the Riemann Hypothesis (RH implies LH, but not vice versa). We also discuss the (very slow) progress made towards the Lindelöf Hypothesis, and in comparison how difficult it makes the Riemann Hypothesis appear. 00:00 - Intro 00:44 - The mu function 02:20 - Growth rate of the Riemann zeta function 04:21 - Case 1 proof: sigma greater than 1 06:23 - Case 2 proof: sigma equals 1 21:21 - Case 3 proof: sigma less than or equal to 0 24:34 - History/Explanation of the mu function and the Lindelöf Hypothesis 27:10 - Table of improvements to mu(1/2) 28:30 - Relation to number of zeros in the critical strip and the Riemann Hypothesis 32:12 - Global bound for the number of zeros on the critical line and the Riemann Hypothesis