A math course explaining the Riemann zeta function
Curated by: Zeta Explained (121 videos)
This is the 95th 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. Sources Titchmarsh, "The Theory of the Riemann Zeta-Function". https://sites.math.rutgers.edu/~zeilberg/EM18/TitchmarshZeta.pdf Ivić, "The Riemann Zeta-Function: Theory and Applications" Graham & Kolesnik, "Van der Corput's Method of Exponential Sums" This particular video discusses the theory of exponent pairs, which arise from van der Corput's A and B processes, and how they pushed the asymptotic growth rate of the zeta function on the critical line lower than 1/6. We specifically walk through how to think about early results in bounding the zeta function as specific exponent pairs. 00:00 - Intro and the Lindelöf Hypothesis 03:41 - Table of progress on bounding the order of zeta on the critical line 05:33 - A and B Rules of exponent pairs 10:58 - The tree of the A and B processes 16:49 - Reframing early results on Lindelöf Hypothesis as exponent pairs 19:00 - Brute force search on the tree 21:37 - Rankin's bound of 0.1645106784 26:18 - Math of exponent pairs 31:09 - Proof of Titchmarsh's 27/164 bound 35:43 - Proof of Phillips's 229/1392 bound 36:34 - Proof of the general zeta bound θ = (1/2)* (p+q-1/2) 37:28 - Summary