Zeta Explained Course

Currently Playing: Zeta Explained #81: Divisor Functions and Dirichlet Series

This is the 81st 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 follow "The Riemann Zeta-Function" by Aleksandar Ivić. This particular video discusses d(n), the function counting the number of divisors of n. We also define ω(n) to be the number of distinct prime factors of n. We mention Wigert's theorem on bounding the growth of d(n). Then we prove the following identities: ζ^2(s)/ζ(2s) = ∑_(n=1 to ∞) 2^ω(n)/n^s ζ^3(s)/ζ(2s) = ∑_(n=1 to ∞) d(n^2)/n^s ζ^4(s)/ζ(2s) = ∑_(n=1 to ∞) d^2(n)/n^s 00:00 - Intro 02:26 - d(n) and ω(n) 05:44 - Bounds on d(n) and ω(n). 08:47 - Multiplicative functions 11:16 - Theorem 2: ζ^2(s)/ζ(2s) = ∑_(n=1 to ∞) 2^ω(n)/n^s 16:54 - Theorem 3: ζ^3(s)/ζ(2s) = ∑_(n=1 to ∞) d(n^2)/n^s 20:45 - Theorem 4: ζ^4(s)/ζ(2s) = ∑_(n=1 to ∞) d^2(n)/n^s 23:00 - Summary