Zeta Explained Course

Currently Playing: Zeta Explained #43: Riemann's Connection Between ζ and Primes

This is the 43rd 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 "Riemann's Zeta Function" by H.M. Edwards. This particular video covers the second half of Riemann's 1859 paper, where he discusses the prime counting functions π(x) and J(x) (using modern notation), a formula for J(x) in terms of π(x), a Möbius inversion for π(x) in terms of J(x), how to write log ζ(s) in terms of J(x), Fourier inversion to get J(x) in terms of log ζ(s), von Mangoldt's proof, an explicit formula for π(x), and the visual interpretation of zeros of the zeta function as cosine waves. Riemann's 1859 paper here: https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-translation.pdf 00:00 - Intro and a graph of Riemann's approximation for π(x) 02:27 - π(x) and J(x) 05:53 - Li(x) 09:34 - Graph of π(x) vs J(x) vs Li(x) 10:15 - Relation between π(x) and J(x) 13:24 - Möbius inversion of π(x) and J(x) 18:14 - Math 24:14 - Fourier inversion of (log ζ(s))/s and J(x) 25:04 - Using the ξ(s) function (from Episode 42) to get Riemann's main equation for J(x) 28:23 - von Mangoldt's method of proving Riemann's main equation for J(x) 33:11 - Cosine waves from the nontrivial zeros of ζ(s) 37:32 - Graphs of adding cosine waves to Riemann's approximation of π(x) 41:57 - Conclusion