Axioms for L-functions (RH Saga S1E4)

This is the fourth episode of the RH Saga

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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.

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Chapters:
00:00 - Intro
02:57 - Analytic continuation
05:25 - Functional equation
14:42 - Euler product
29:37 - Temperedness
30:58 - Final remarks

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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

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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.

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Social:

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