A math course explaining the Riemann zeta function
Curated by: Zeta Explained (121 videos)
This is the 82nd 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 a proof of Ramanujan's formula relating the Riemann zeta function to the generalized divisor function. In particular, ζ(s)ζ(s-a)ζ(s-b)ζ(s-a-b)/ζ(2s-a-b) = ∑_(n=1 to ∞) σ_a(n)σ_b(n)/n^s We'll start with the simpler result that ζ(s)ζ(s-a) = ∑_(n=1 to ∞) σ_a(n)/n^s 00:00 - Intro 02:37 - σ_a(n) 03:57 - Dirichlet series algebra: ζ(s)ζ(s-a) = ∑_(n=1 to ∞) σ_a(n)/n^s 09:26 - Proof that σ_a(n) is multiplicative 13:37 - Proof of Ramanujan's formula