Zeta Explained Course

Currently Playing: Zeta Explained #16: Proof of Riemann's Functional Equation

This is the 16th 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 proves Riemann's functional equation using a classic approach from Alexandar Ivić. We use Stieltjes integration, Fourier series, a Mellin transform, some identities of the Gamma function, and the Dominated Convergence Theorem. 00:00 - Intro 01:35 - Start of proof, writing ζ(s) as a Stieltjes integral involving a sawtooth function 10:15 - Substituting the sawtooth function with its Fourier series 15:33 - Deriving an asymmetric form of Riemann's functional equation 17:31 - Turning the asymmetric form into the symmetric form 20:27 - Justifying the interchange of infinite integral and infinite sum, using the Dominated Convergence Theorem