Nina Snaith Random matrices and Riemann zeros April 2010 Research supported by: TexPoint fonts used in EMF: A A A A A.

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Presentation transcript:

Nina Snaith Random matrices and Riemann zeros April 2010 Research supported by: TexPoint fonts used in EMF: A A A A A

2 Riemann zeta function Re s Im s 01/2 and -21

3

From: M.L.Mehta, Random Matrices, 3 rd Ed, Elsevier, 2004

5 Random Unitary Matrices

6 First zeros

7 around the th zero

8 Picture by A. Odlyzko 79 million zeros around the th zero Mystery 0: why are these statistics the same?!

Hugh Montgomery: Mathematician Freeman Dyson: Physicist In 1972 these two met:

11 Distribution of log ζ (Selberg) :

12 Numerical distribution (due to Odlyzko) of values of around the height of the th zero compared to the standard Gaussian Mystery 1: why is the data so far from the Gaussian?

13

14 The value distribution for vs the real part of the log of the Riemann zeta function Keating and Snaith (2000)

15 Mystery 2: what is the true size of ζ?

16 Hybrid formula, Gonek, Hughes and Keating (2007)

17

18