Zeta Explained #22: The Classical Zero-Free Region Inside the Critical Strip скачать в хорошем качестве

Zeta Explained #22: The Classical Zero-Free Region Inside the Critical Strip

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.)