Zeta Explained Course

Currently Playing: Zeta Explained #100: Riemann Hypothesis Implies Lindelöf Hypothesis

This is the 100th 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 Theorem 14.2 of Tichmarsh's text, which is a proof that the Riemann Hypothesis implies the Lindelöf Hypothesis. The Riemann Hypothesis is an unproven conjecture on the location of the zeros of the zeta function, namely that all zeros are the the critical line Re(s)=1/2. Whereas, the Lindelöf Hypothesis is an unproven conjecture about the growth rate of zeta on the critical line, namely that ζ(1/2+it) = O(t^ε). 00:00 - Intro 03:40 - Proof sketch 06:42 - Proving that log |ζ(2+it)| = O(1) 11:09 - 3 complex analysis lemmas 16:40 - Proof 38:16 - Outro