Zeta Explained Course

Currently Playing: Zeta Explained #85: Landau's Theorem on Dirichlet Series

This is the 85th 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 Landau's theorem on Dirichlet series with nonnegative coefficients. Specifically, let F(s) = ∑_(n=1 to ∞) f(n)/n^s be a Dirichlet series, where f(n) ≥ 0 for all n. Let σ be the abscissa of convergence. Then F(s) either converges everywhere or has a singularity at the real point σ. We prove this theorem, and also discuss a case where f(n) can be negative and why the theorem doesn't work. 00:00 - Intro 03:35 - Diagram of convergence regions, statement of theorem 05:58 - Proof of Landau's theorem 16:36 - What happens if f(n) can be negative? 1/ζ(s) and the Riemann hypothesis. 22:16 - Summary