Zeta Explained Course

Currently Playing: Zeta Explained #45: Von Mangoldt's Explicit Formula for ψ(x)

This is the 45th 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 von Mangoldt's explicit formula for Chebyshev's prime counting function ψ (psi) function. We discuss Riemann's prime counting function J(x), why it was difficult to prove, and why mathematicians found an alternative proof. We go over a proof of von Mangoldt's explicit formula via an expanding contour integral. 00:00 - Zeros of the zeta function 03:52 - Graphs of J(x) and ψ(x) 05:54 - Proof outline 11:32 - Diagram of the proof 13:10 - Proof via contour integration 41:51 - Connecting the formula for ψ(x) to prime waves 43:14 - Graph approximating ψ(x) using partial sums of von Mangoldt's explicit formula