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

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

3. 3

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

5. 5 Random Unitary Matrices

6. 6 First 100000 zeros

7. 7 around the th zero

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

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

11. 11 Distribution of log ζ (Selberg) :

12. 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?

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

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

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

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