Zeta Explained Course

Currently Playing: Zeta Explained #11: The Laurent Series of the Zeta Function at s=1

This is the 11th 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 the Laurent series and finding the coefficients for the zeta function at the singularity s=1. 00:00 - What is a Laurent series 01:30 - Defining principal part vs analytic part 02:00 - Quick crash course on complex analysis 03:54 - Start of finding the Laurent series for ζ(s) at s=1, in particular the residue term 05:45 - Formula for the coefficients γ_k of the rest of the terms, including the Euler-Mascheroni constant γ 08:08 - Proof of the formula for the coefficients γ_k. Start by defining auxiliary constant c_r 11:04 - Visual intuition for what integrating against the sawtooth function means 16:43 - Proof continued. More auxiliary functions: S_r(x) 24:29 - Proof sketch. Will prove a left-hand-side and a right-hand-side 26:38 - Proof of left-hand-side 37:36 - Proof of right-hand-side 44:13 - Combining the left-hand-side and right-hand-side