A math course explaining the Riemann zeta function
Curated by: Zeta Explained (121 videos)
This is the 80th 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 discusses Pólya's conjecture concerning the Liouville function λ(n), which is equal to +1 if n has an even number of prime factors and -1 if n has an odd number of prime factors, with multiplicity. For example, 12 = 2 * 2 * 3 is considered t have 3 prime factors in this definition, and thus λ(12) = -1. Pólya's conjecture is that L(x) defined as ∑_(n≤x) λ(n) is never positive starting at x=2. We show using the locations of the zeros of the Riemann zeta function why the L(x) function has the shape it does (negative with oscillations), and mention that the oscillations eventually align to make L(x) briefly positive starting at x=906,150,257 (Tanaka 1980), thus falsifying Pólya's conjecture. 00:00 - Intro 03:18 - Naive graphs of L(x) 05:41 - Perron's inversion formula 14:22 - An explicit formula for approximating L(x) 19:36 - Graphs of L(x) with the explicit formula 22:35 - Rephrasing Pólya's conjecture as a wave interference question