Zeta Explained Course

Currently Playing: Zeta Explained #10: Euler–Maclaurin Summation

This is the tenth 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 analytic continuation of zeta using Euler–Maclaurin summation. 00:00 - Intro 02:06 - Plan for the video 02:22 - Start of derivation of the Euler-Maclaurin summation formula 08:02 - First iteration (first integration by parts) 10:29 - Second iteration (second integration by parts) 12:20 - Bernoulli polynomials 16:04 - The Euler-Maclaurin summation formula after two iterations 18:55 - The Euler-Maclaurin summation formula after 2n iterations 20:34 - Computing the specific formula for zeta 26:38 - The specific formula for zeta 27:16 - The formula can converge at arbitrary points