Zeta Explained Course

Currently Playing: Zeta Explained #54: Littlewood's Log Log Log Bound on Zeta Gaps

This is the 54th 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 Littlewood's theorem that the maximum gap between zeta zeros is bounded by O(1/(log log log T)) as T goes to infinity, though we'll follow Titchmarsh's proof, not Littlewood's. We start with Ep 53's classical bound of 12 between between successive zeros of the Riemann zeta function for a large enough height T. The idea is to use the Borel–Carathéodory Theorem and the Hadamard Three-Circle Theorem repeatedly in a chain of circles. 00:00 - Background and statement of the theorem 08:01 - Lemma 1: Maximum Modulus Principle 10:45 - Lemma 2: Borel–Carathéodory Theorem 12:30 - Lemma 3: Hadamard Three-Circle Theorem 14:32 - log ζ(s) is well-defined 16:11 - Proof of Littlewood's theorem 38:35 - Conclusion and corollary