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
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?
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)