Zeta Explained Course

Currently Playing: Zeta Explained #33: The Mertens Conjecture (and the Riemann Hypothesis)

This is the 33rd 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 discusses the Möbius function μ(n), the Mertens function M(x), the Mertens conjecture, how the Mertens conjecture implies the Riemann hypothesis, and a weaker form of the Mertens function growth rate which (if true) would still imply Riemann hypothesis. We will follow the book "Riemann's Zeta Function" by H. M. Edwards. We also explain a result from the paper "Computations of the Mertens Function and Improved Bounds on the Mertens Conjecture" (Hurst 2016) 00:00 - The Möbius function μ(n) and the Mertens function M(x) 02:53 - Plots of M(x) 08:03 - Background of Mertens Conjecture 13:02 - Riemann zeta function (and Möbius inversion) 14:00 - Proof that Mertens conjecture implies Riemann Hypothesis (of course, Mertens conjecture is false, so this doesn't prove RH) 23:53 - Weaker form of Mertens function growth rate would still imply RH. 26:28 - MC implies RH, but weaker form is equivalent to RH 29:04 - Random walk theory 31:03 - Comparison of statements about the order of M(x) 34:25 - Absurdities of the Mertens function: M(∞) = -2 ???