Zeta Explained Course

Currently Playing: Zeta Explained #34: How the Mertens Conjecture was Disproved

This is the 34th 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. This particular video walks through Odlyzko and te Riele's famous disproof of the Mertens Conjecture from 1985. We will cover various topics such as: the Möbius function, the Riemann zeta function, the Riemann Hypothesis, Perron's inversion formula, contour integration, Kronecker's theorem of simultaneous diophantine approximations, Ingham's method, and the Lenstra–Lenstra–Lovász lattice reduction algorithm. "Disproof of the Mertens Conjecture" (Odlyzko and te Riele 1985) https://www-users.cse.umn.edu/~odlyzko/doc/arch/mertens.disproof.pdf "Computations of the Mertens Function and Improved Bounds on the Mertens Conjecture" (Hurst 2016) https://arxiv.org/abs/1610.08551 00:00 - Intro 04:26 - Start following the paper 11:46 - Some "elementary" math (how does Mertens Conjecture relate to Riemann Hypothesis) 17:29 - How the authors got a very complicated looking formula for M(x), i.e. equation (2.3) 27:51 - How to interpret equation (2.3) 34:06 - How cosines involving the zeros of the zeta function show up 41:00 - Visualizing what these cosines mean 47:30 - Discussion of how if the zeros of the zeta function had linearly independent imaginary parts, then Kronecker's theorem immediately disproves Mertens Conjecture 52:05 - How to proceed without assuming linear independence 55:42 - Overview of Ingham's method 1:02:20 - How finding where cosines are close to 1 relates to finding when a line is close to a lattice 1:06:26 - The Lenstra–Lenstra–Lovász lattice reduction algorithm 1:13:08 - The spike of cosines that disproves Mertens conjecture