Zeta Explained Course

Currently Playing: Zeta Explained #93: van der Corput's B Process

This is the 93rd 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 book "The Theory of the Riemann Zeta-Function" by E.C. Titchmarsh. https://sites.math.rutgers.edu/~zeilberg/EM18/TitchmarshZeta.pdf We also use Ivić's text, "The Riemann Zeta-Function: Theory and Applications". This particular video continues the theory of exponential sums by van der Corput, specifically the B process. The main idea is to approximate exponential sum with a different sum of exponential integrals. We show a simple example of swapping a sum into a different sum, and then go through four technical lemmas/theorems listed in the timestamps below. 00:00 - Intro and example of swapping a sum into another sum 03:03 - Outline 04:16 - Lemma 4.7. Changing a sum into a sum of exponential integrals 11:11 - Lemma 4.6. Computing an exponential integral 19:36 - Theorem 4.9. The van der Corput B process dual sum 22:26 - Theorem 5.9. A simple van der Corput B process bound