Zeta Explained Course

Currently Playing: Zeta Explained #50: Littlewood's Log Log Log Bound on the Prime Number Theorem

This is the 50th 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. This particular video covers Littlewood's famous 1914 result, the Ω(sqrt(x) log log log x) bound on the error term of the Prime Number Theorem. We walk through the proof showing that that (psi(x)-x)/sqrt(x) can be written as an infinite sum of sine waves, and must be unbounded using Dirichlet's approximation theorem. Keeping careful track of the unboundedness results in the Ω(log log log x) for a finite sum. Then we see that the remaining infinite sum cannot exceed the finite sum, thus proving the infinite sum is also Ω(log log log x). 00:00 - Prime counting functions and error bounds 11:28 - Proof pt 0: Trivial case 13:51 - Proof pt 1: (Interesting) Getting a finite sum to exceed C log log log x using Dirichlet's approximation theorem 44:48 - Proof pt 2: (Uninteresting) Dealing with the tail sum using an exponential damping function 1:01:36 - Final remarks