Zeta Explained Course

Currently Playing: Zeta Explained #91: The Hardy–Littlewood 1/6 Bound on the Lindelöf Hypothesis

This is the 91st 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 "The Theory of the Riemann Zeta-Function" by E.C. Titchmarsh. https://sites.math.rutgers.edu/~zeilberg/EM18/TitchmarshZeta.pdf This particular video covers the Hardy–Littlewood result that ζ(1/2+it) = O(t^(1/6+ε)). We start by stating the Lindelöf hypothesis and a table of historical results over the past century, starting with Lindelöf's 1/4 bound, and how Hardy and Littlewood improved this down to 1/6. 00:00 - Intro 08:10 - Proof sketch 10:02 - Proof part 1 (partial summation and Weyl differencing) 17:54 - Visualization of proof part 1 21:21 - Proof part 2 (dealing with the cosecant) 28:44 - Visualization of proof part 2 34:07 - Proof part 3 (remaining math and dyadic intervals)