Zeta Explained Course

Currently Playing: Zeta Explained #56: S(T) as a Divergent Sum of Prime Waves

This is the 56th 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. We follow the text "The Theory of the Riemann Zeta-Function" by E.C. Titchmarsh. This particular video follows up discussion of the S(T) function that we defined in Ep 55. S(T) serves as the sawtooth-like error term in the Riemann–von Mangoldt formula. We show that S(T) can be thought of as a divergent sum of sine waves where the frequencies are the logs of prime powers. 00:00 - Background 07:20 - Math showing S(T) as sine waves 16:00 - Visualization of divergent sine waves 19:13 - Smoothing a cutoff 22:05 - Selberg smoothing 28:44 - Bonus 1: Proving Littlewood's theorem that under the Riemann Hypothesis, S(T) = O(log T/log log T) 31:31 - Bonus 2: Duality between zeta zeros and primes