Zeta Explained #42: Riemann's ξ Function and the Origin of the Riemann Hypothesis
Автор: Zeta Explained
Загружено: 2026-03-09
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Описание:
This is the 42nd 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. We follow the book "Riemann's Zeta Function" by H.M. Edwards.
This particular video covers three paragraphs in Riemann's 1859 paper, including where the Riemann Hypothesis is stated. We cover the definition of the xi function (ξ), Riemann's rotated definition, the zero free region, the Riemann–von Mangoldt Formula, the 100% Hypothesis, the Riemann Hypothesis, the Weierstrass and Hadamard Factorization Theorems, and Euler's factorization of sine.
Riemann's 1859 paper here: https://www.claymath.org/wp-content/u...
00:00 - Intro and defining ξ
06:16 - Riemann's version of ξ which is rotated 90 degrees
14:04 - No zeros in Re(s)＞1 or Re(s)＜0
17:23 - Graph of zero-free region
18:56 - The Riemann–von Mangoldt Formula
23:10 - The 100% Hypothesis and the Riemann Hypothesis
30:22 - Writing ξ in terms of its zeros
42:37 - Summary
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