Zeta Explained #17: Riemann's Xi Function скачать в хорошем качестве

Zeta Explained #17: Riemann's Xi Function

This is the 17th 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 O(T log T). 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 covers Riemann's xi function and its relation to the Riemann zeta function, motivation for the function, its symmetric property, and some history. 00:00 - Intro 02:24 - The problems with the zeta function 03:02 - How to remove a pole 05:19 - Removing the poles and trivial zeros from the zeta function, and call the new function "xi" 08:38 - Deriving the symmetrical form of the xi function 11:15 - Properties of the xi function 12:59 - Historical notation for the xi function