Zeta Explained #15: Riemann's Functional Equation скачать в хорошем качестве

Zeta Explained #15: Riemann's Functional Equation

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 the Riemann-Von Mangoldt equation estimating the number of zeros in the critical strip between 0 and T as O(T log T). 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 Alexandar 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