Zeta Explained Course

Currently Playing: Zeta Explained #15: Riemann's Functional Equation Connects ζ(s) and ζ(1-s)

This is the 15th 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 Riemann's functional equation, the trivial zeros of the zeta function, and conjugate symmetry. 00:00 - Intro (Riemann's 1859 paper) 00:42 - Riemann's functional equation 01:12 - Computing ζ(-1) = -1/12 03:57 - 1+2+3+4+... = -1/12? 05:20 - ζ(s) = 0 when s is a negative even integer 07:29 - ζ(0) = -1/2 09:21 - Using the functional equation on complex numbers, the symmetry across the line "Re(s) equals 1/2" 10:13 - Conjugate symmetry 13:49 - Combining conjugate symmetry with the functional equation symmetry 16:09 - No zeros when "Re(s) is at most 0" other than the negative even integers 17:22 - The critical line and the critical strip 18:26 - Zeros in the critical strip have 2 or 4 copies (depending on if they're on the line Re(s) = 1/2) 19:16 - Recap