Zeta Explained #22: The Classical Zero-Free Region Inside the Critical Strip
Автор: ZetaExplained
Загружено: 2026-02-03
Просмотров: 26
Описание:
This is the 22nd 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 T/(2π)log(T/(2π)) - T/(2π). 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 introduces and proves a classical result from de la Vallée Poussin that there are no zeros in the region σ greater than 1-c/(log t)
00:00 - Intro
02:04 - Graph with symmetries
02:39 - Theorem and proof sketch
04:40 - Proof part 1: Real part of ζ'(s)/ζ(s) using cosines
09:06 - Proof part 2: Bound on real part of ζ'(s)/ζ(s) using Hadamard factorization
11:40 - Proof part 3: Analyzing ζ'(s)/ζ(s) near 3 points
17:38 - Proof part 4: Finding the bound
23:14 - Final remarks
(23:16 - Sorry, I said "converse" but meant "contrapositive", i.e. we proved if sigma+it IS a zero, then it must be that sigma is less than 1-c/(log t); therefore, if sigma is at least 1-c/(log t), sigma+it must NOT be a zero.)
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