Zeta Explained Course

Currently Playing: Zeta Explained #12: The Explicit Formula for Zeta at Even Integers

This is the 12th 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 covers Euler's explicit formula for zeta(n) at even integers, goes over its proof, and also discusses what is known of zeta at odd numbers. 00:00 - Zeta at positive even integers 01:14 - Discussion of zeta at positive odd integers 03:15 - Euler's formula for zeta at positive even integers 05:59 - Proof sketch of Euler's formula 08:29 - Using the factorization of the sine function 13:07 - Comparing terms of the generating function in two different ways 18:21 - Wrapping up the proof 19:34 - Discussion of non-positive even integers