Zeta Explained Course

Currently Playing: Zeta Explained #49: An Omega Bound on the Prime Number Theorem

This is the 49th 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 Riemann Zeta-Function: Theory and Applications" by Aleksandar Ivić. This particular video covers a classical omega bound on the error term of the Prime Number Theorem. We show that assuming ψ(x) = x + o(x^{1/2}) leads to a contradiction involving a nontrivial zero of the zeta function, and thus, ψ(x) = x + Ω(x^{1/2}). 00:00 - Intro 05:38 - A clarification on the definition of Ω(x) 07:13 - Proof by contradiction 26:09 - Graphing the Omega bound 34:51 - Some more theory of Omega bounds, the (log log log x) bound