Zeta Explained #08: The Dirichlet Eta Function

This is the eighth 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 the Dirichlet eta function. 00:00 - Intro 00:27 - Alternating series (the eta function) 01:52 - Algebra on the alternating series to write it in terms of ζ(s) 05:35 - Convergence of the eta function 08:06 - Geometric intuition for why it converges 13:48 - More rigorous proof why it converges, using Taylor series 18:25 - Recap of the series 18:59 - How the singularities of the first team lead to zeros of the eta function. 23:37 - Continued discussion of zeta