Zeta Explained #76: Euler's Totient Function and its Dirichlet Series

This is the 76th 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. Today we (loosely) follow the text "The Riemann Zeta-Function" by Aleksandar Ivić. This particular video covers some lemmas and results involving the Euler totient function φ(n). We prove some lemmas and then show 3 proofs that the Dirichlet series is ∑_(n=1 to ∞) φ(n)/n^s = ζ(s-1)/ζ(s). 00:00 - Intro 03:21 - Lemma 1: n = ∑_(d|n) φ(d) 06:12 - Lemma 2: φ(n) = ∑_(d|n) μ(d) n/d 08:28 - Lemma 3: φ(n) = n Π_(p|n) (1-1/p) 11:47 - Thm 1 Proof 1: Brute force plugging into the series 17:43 - Thm 1 Proof 2: Dirichlet convolution 22:21 - Thm 1 Proof 3: Generalized Euler product formula 29:18 - Application of Thm 1, using Perron's inversion formula 34:04 - Summary of 3 proofs