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Axioms for L-functions (RH Saga S1E4)

This is the fourth episode of the RH Saga. We finally give a rigorous definition of what an L-function actually is, using four axioms. The axioms include the functional equation (encoding the fundamental symmetry inherent in every L-function) and the Euler product (encoding the connection between L-functions and the primes). The overall aim of RH Saga Season 1 is to map the landscape of L-functions, as a foundation for in-depth exploration of some of the most immortal math problems of all time. This video is part of a PeakMath course. Join the journey at https://www.peakmath.org/ --- Chapters: 00:00 - Intro 02:57 - Analytic continuation 05:25 - Functional equation 14:42 - Euler product 29:37 - Temperedness 30:58 - Final remarks --- Links: 1. LMFDB: https://www.lmfdb.org/ 2. LMFDB History of L-functions: https://www.lmfdb.org/knowledge/show/... 3. Paper by Farmer, Pitale, Ryan and Schmidt on L-function axioms: https://arxiv.org/pdf/1711.10375.pdf --- Errata: 1:33 “Millions of examples”: To clarify, there are infinitely many L-functions, so a lot more than mere “millions”. The point here was to say that many unproven properties of L-functions (like the RH up to some height in the critical strip) have been checked in millions of examples. --- Social: https://www.peakmath.org/ #RiemannHypothesis #F1Geometry #Mathematics #PeakMath #RHSaga #Langlands