Zeta Explained Course

Currently Playing: 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 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 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.)