Riemann’s Hypothesis

R.S.MacKay, University of Warwick

NEW TIME! Tuesdays 12.10-14.00, 10 Oct - 5 Dec (minus 31 Oct)

Disclaimer: I am not trained in this area, but the problem has resisted solution for over 150 years so I thought it might benefit from a view from a naive position and I became obsessed with it. The goal of the course is to get you interested in trying out ideas. There will be a lot of interesting mathematics whatever.

1. Statement, Significance & Numerical evidence

2. Outline of some possible approaches (no positive local minima, Fourier transform, radial Brownian walks, spectral, …)

3. More depth on Spectral approaches: (a) 1D Schrodinger (b) Magnetic Laplacian on a hyperbolic surface (c) Zagier’s Laplacian on a hyperbolic surface (d) Multi-component Schrodinger operators (e) Dirac systems (f) Linear Hamiltonian systems

4. Dirichlet L-functions

References

The best starting point is: Riemann GFB, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monat. der Königl. Preuss. Akad. der Wissen. zu Berlin aus der Jahre 1859 (1860) 671-680; or translation into English by Wilkins DR (1998)

A nice introduction to the subject is Bombieri's statement of the Clay Math problem.

A resource that collects together lots of relevant papers (including the above) is: Borwein P, Choi S, Rooney B, Weirathmueller A, The Riemann Hypothesis (Springer, 2008)

A reference for part of the module is: RS MacKay, Towards a spectral proof of Riemann’s hypothesis, arxiv

Chasing citation trails and googling will find you plenty more.

Here is a Mathematica script to illustrate some things. I want to use this during class but it seems the technology is not up to it, so best if you try on your own! Alternatively, here is a pdf file of the output.

Assessment: I will ask each of you who would like to be assessed to present a short oral or written presentation on some aspect of the topic.