Zeta Explained Course

Currently Playing: Zeta Explained #79: Liouville's Function

This is the 79th 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 discusses the Liouville function λ(n), which is equal to +1 if n has an even number of prime factors and -1 if n has an odd number of prime factors, with multiplicity. For example, 12 = 2 * 2 * 3 is considered t have 3 prime factors in this definition, and thus λ(12) = -1. We cover its relation to whether more numbers have an odd vs even number of prime factors. And then prove the Dirichlet series involving the Riemann zeta function: ζ(2s)/ζ(s). 00:00 - Intro 06:23 - Theorem 1: ∑_(n=1 to ∞) λ(n)/n^s = ζ(2s)/ζ(s) 13:23 - Perron's inversion formula on Theorem 1 22:02 - Table of λ(n) and L(x) 24:19 - Graphs of L(x)