Nina Snaith

Random matrices and Riemann zeros

April 2010

Research supported by:

2 Riemann zeta function

Re s

Im s

0 1/2 and-2 1

3

From: M.L.Mehta,

Random Matrices, 3rd Ed,

Elsevier, 2004

5 Random Unitary Matrices

6

First 100000

zeros

7

zeros

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)

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

16

Hybrid formula, Gonek, Hughes and Keating (2007)

17

18