Zeta Explained Course

Currently Playing: Zeta Explained #83: Using Zeta to Solve Arithmetic Identities

This is the 83rd 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. This particular video covers an algebraic approach of thinking about Dirichlet series using Dirichlet convolution. We cover the Dirichlet convolution formula: (f * g)(n) = ∑_(d|n) f(d)g(n/d) and use it to prove the following identities: ∑_(d|n) μ(d) = 0 (where μ(n) is the Möbius function) ∑_(d|n) φ(d) = n (where φ(n) is the Euler totient function) ∑_(d|n) Λ(d) = log n (where Λ(n) is the von Mangoldt function) For example, the first identity with the Möbius function can be reduced to the trivial-sounding statement of ζ(s) * (1/ζ(s)) = 1. 00:00 - Intro 02:00 - Dirichlet series 13:55 - Solving arithmetic problems