Zeta Explained Course

Currently Playing: Zeta Explained #94: Alternate Proof of 1/6 Bound on Lindelöf Hypothesis (van der Corput's Method)

This is the 94th 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 an alternate proof of 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. We'll compare how Hardy and Littlewood's diverges from the proof using van der Corput's method. Instead of using a process that approximates the "log n" term with a truncated Taylor series, we can deal with the log directly. The final result is very, very slightly stronger than the Hardy–Littlewood result, in that we go from O(t^(1/6) log^(3/2) t) to O(t^(1/6) log t) which reduces the power of the log. 00:00 - Intro 04:03 - Table of historical bounds 04:34 - Proof sketch comparing the Hardy–Littlewood to the new proof which uses the method of van der Corput 07:00 - Start of proof 09:15 - van der Corput's A and B processes and Theorem 5.11 which bounds a sum using the third derivative 14:20 - Resume proof 22:42 - Teaser for Ep 95