Zeta Explained Course

Currently Playing: Zeta Explained #35: Diophantine Approximation

**Error at 43:46** I said we can use Kronecker's theorem with some complication of linear independence, but this is misleading. Since the cosines here have no offset, we don't have to use Kronecker's theorem--we can just use Dirichlet's theorem, which doesn't care about linear independence. No offset means cos(a_1 x), cos(a_2 x), ..., as opposed to cos(a_1 x + b_1), cos(a_2 x + b_2), ... This is the 35th 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. The viewer is expected to understand calculus and complex numbers, whereas I will try to explain concepts from complex analysis as needed. We will follow the book "The Riemann Zeta Function: Theory and Applications" by Aleksandar Ivić. This particular video introduces the the concept of Diophantine approximation, and we cover two theorems: Dirichlet's approximation theorem and Kronecker's approximation theorem. 00:00 - Diophantine equations 04:39 - Dirichlet's approximation theorem 14:30 - Visualizing Dirichlet's approximation theorem 19:20 - Kronecker's approximation theorem 23:14 - Visualizing Kronecker's theorem on a square 27:05 - Visualizing Kronecker's theorem on a torus 32:32 - Kronecker's theorem on waves 37:45 - Visualizing Kronecker's theorem on waves 39:57 - Relation to the Mertens conjecture 42:30 - Relation to the Riemann zeta function