Zeta Explained Course

Currently Playing: Zeta Explained #53: Zeta Gaps (Are ≤12)

This is the 53rd 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 shows a classical constant bound of 12 between between successive zeros of the Riemann zeta function for a large enough height T. We use the Borel–Carathéodory Theorem and the Hadamard Three-Circle Theorem. We also mention the Riemann–von Mangoldt equation which in some sense supercedes this result, thought the result proves that there cannot exist pathological spacings of zeros where the are large clumps of zeros followed by large gaps. Next time, we'll prove that the bound actually shrinks like 1/(log log log T) as we go to higher imaginary values T. 00:00 - Background for distribution of nontrivial zeros 05:55 - Proof sketch and elementary theory 13:05 - Complex log 23:24 - Lemma 1: Maximum Modulus Principle 26:22 - Lemma 2: Borel–Carathéodory Theorem 28:35 - Lemma 3: Hadamard Three-Circle Theorem 31:07 - Proof 43:07 - Result and Corollaries