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

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