Zeta Explained Course

Currently Playing: Zeta Explained #51: Skewes' Number and Gauss's False Conjecture on Primes

This is the 51st 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 text "Riemann's Zeta Function" by H.M. Edwards. This particular video covers Littlewood's famous 1914 result, the Ω(sqrt(x) log log log x) bound on ψ(x)-x, and how this result disproves a conjecture made by Gauss that π(x)＜Li(x). We also cover some history behind this result. 00:00 - Intro and history of Gauss's conjecture 11:14 - The four prime counting functions 16:18 - Riemann's Möbius inversion: J(x) to π(x) 23:09 - Translating ψ(x) to J(x) 26:29 - Analyzing Littlewood's error bound to disprove Gauss's conjecture 33:26 - The conclusion of the proof (why Gauss's conjecture is false) 38:09 - Intuition behind why there is a log log log, and why Skewes' number is so large