Zeta Explained Course

Currently Playing: Zeta Explained #19: Proof that the Number of Zeros in the Critical Strip Is O(T log T)

This is the 19th 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 the growth rates of the Riemann xi function, and then uses Jensen's formula to find that the number of zeros in the critical strip up to an imaginary part of T is O(T log T). 00:00 - Intro 01:46 - Proof sketch 03:06 - Start of proof 03:49 - Stirling approximation on the Gamma function 13:37 - Stieltjes integral for the Riemann zeta function 15:16 - Growth rate of the Riemann xi function 17:19 - Jensen's formula 19:58 - Proof continued 26:32 - Conclusion and recap