Zeta Explained #20: Hadamard's Factorization of the Zeta Function
This is the 20th 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 Weierstrass Factorization Theorem and Hadamard Factorization Theorem, and uses these theorems to find a factorization for the Riemann xi and zeta functions. The result was first found by Hadamard in 1893. It is also known as the Hadamard product formula.
00:00 - Intro
03:13 - Prerequisite knowledge
06:02 - Weierstrass Factorization Theorem
12:33 - Explanation of why exponential terms are needed in Weierstrass Factorization Theorem
23:16 - Motivation for how many exponential terms are needed
28:41 - Hadamard Factorization Theorem
30:16 - Solving for the constants in the Hadamard factorization
37:03 - Final result (Hadamard Product Formula for the Riemann zeta function)