A math course explaining the Riemann zeta function
Curated by: Zeta Explained (121 videos)
This is the 75th 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 text "Introduction to Analytic Number Theory" by Tom M. Apostol. This particular video covers the visible grid point problem (or visible lattice point problem). We go over two proofs that the probability that a random grid point is visible from the origin is 6/π^2 = 1/ζ(2). These proofs involve number theory concepts like "relatively prime" and Euler's totient function. We also go through the problem in n-dimensions, and find that the probability is 1/ζ(n). 00:00 - Intro and problem statement 01:18 - Two lemmas on gcd 02:54 - First proof of 6/π^2 08:43 - Euler's totient function φ(n) and three lemmas 18:33 - Second proof of 6/π^2 26:33 - The probability in n-dimensions is 1/ζ(n)