This is the sixth episode of the RH Saga.
Continuing from the last episode, we now consider "invariants", i.e. machines taking L-functions as input and giving a number as output. These invariants are intimately connected to several Millennium Problems.
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 www.peakmath.org/
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Chapters:
00:00 - Intro
03:16 - First examples
11:36 - Keiper-Li coefficients
18:04 - Structure
26:42 - Millennium problems
32:26 - Duality
36:02 - Final remarks
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Links:
1. LMFDB:
www.lmfdb.org/
2. LMFDB, Ramanujan Delta:
www.lmfdb.org/L/2/1/1.1/c11/0/0
3. LMFDB, Elliptic curve conductors less than 100:
www.lmfdb.org/EllipticCurve/Q...
4. "Riemann's Hypothesis" - Brian Conrey:
tinyurl.com/conrey90
5. SageMathCell permalink for Keiper-Li coefficients of K:
sagecell.sagemath.org/?z=eJxd...
6. Fungrim entry on Keiper-Li coefficients:
fungrim.org/topic/Keiper-Li_c...
7. Paper by Farmer, Pitale, Ryan and Schmidt on L-function axioms:
arxiv.org/pdf/1711.10375.pdf
8. Andrew Droll thesis pdf:
qspace.library.queensu.ca/ser...
9. Donald Yau, "Lambda-rings":
www.worldscientific.com/world...
10. Lindsay N. Childs, "A Concrete Introduction to Higher Algebra", Second edition:
link.springer.com/book/10.100...
(There is also a new third edition, which may be less expensive)
11. Mícheál Ó Searcóid, "Elements of Abstract Analysis":
www.amazon.com/Elements-Abstr...
12. Allen Hatcher, "Topology of Numbers":
pi.math.cornell.edu/~hatcher/...
13. The Millennium Problems:
www.claymath.org/millennium-p...
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Errata:
No errors found yet.
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Social:
www.peakmath.org/
#RiemannHypothesis #F1Geometry #Mathematics #PeakMath #RHSaga #Langlands
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