Zeta Explained #99: The Gauss Circle Problem

This is the 99th 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. Sources Graham & Kolesnik, "Van der Corput's Method of Exponential Sums" Ivić, "The Riemann Zeta-Function: Theory and Applications" This particular video discusses the Gauss Circle Problem. Let F(r) be the number of integer lattice points inside a circle of radius r. Since the area of a circle is πr^2, we have F(r) = πr^2 + R(x) where R(x) is an error term. It is an open question as to the growth rate of Δ(x). 00:00 - Intro 02:07 - Animation of the error term 02:47 - Gauss circumference argument 05:12 - Graphs of N(x) and R(x) 07:17 - Growth rate of R(x) 09:14 - r(n) and the sum of two squares 22:16 - Hyperbola argument 25:01 - Review of Dirichlet Divisor Problem 31:30 - Dirichlet series for d(n) and r(n) 40:37 - Connection with the Riemann zeta function and the Lindelöf Hypothesis