Zeta Explained Course

Currently Playing: Zeta Explained #55: S(T) and the Accuracy of the Riemann–von Mangoldt Formula

This is the 55th 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 covers the Riemann–von Mangoldt formula in more detail than before in Ep 21. We show how accurate it is, even at large values of T such as 10^12, 10^14, 10^16, where the formula is off by less than 1. To investigate the accuracy of this formula, we do a contour integral and show that N(T) = L(T) + S(T) + O(1/T), where L(T) contains the main estimation terms, and S(T) is a sawtooth error term. We then prove Littlewood's theorem that even though S(T) = O(log T), the *integral* of S(T) is also O(log T), suggesting that S(T) has a very large degree of cancellation in its oscillation. This provides some intuition for why the Riemann–von Mangoldt is so accurate. 00:00 - Background and super high accuracy 08:01 - Proof deriving the main terms of the Riemann–von Mangoldt formula 16:52 - Splitting out N(T) = L(T) + S(T) + O(1/T) 26:12 - Defining S_1(T) as the integral of S(T) 30:47 - Proving Littlewood's theorem that S_1(T) = O(log T) 45:26 - Conclusion