Zeta Explained Course

Currently Playing: Zeta Explained #09: Riemann–Stieltjes Integration

This is the nineth 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. We will follow the book "The Riemann Zeta Function: Theory and Applications" by Aleksandar Ivić. This particular video covers analytic continuation of zeta using Riemann–Stieltjes integration. 00:00 - The Riemann-Stieltjes integral 01:30 - Comparing to a normal Riemann integral 04:03 - The floor function 05:27 - Integrating with respect to the floor function (why it's just a discrete sum) 09:26 - What we can do with a Stieltjes integral (moving into the integrand, integration by parts) 12:19 - Writing the infinite series of the zeta function as a Stieltjes integral, then integrating by parts 17:32 - Integral formula for ζ(s) converging for "Re(s) greater than 0"