Summer School on The Riemann Zeta Function and Random Matrix Theory
Oberwolfach, October 15 - 21, 2000
Jon Keating (Bristol)
Zeev Rudnick (Tel Aviv)
Nina Snaith (Bristol)
The Riemann zeta function and its generalizations are among the most
useful tools in Number Theory. Random Matrix Theory is a theory of the
local statistics of the eigenvalues of certain ensembles of random
matrices, such as the group of all N-by-N unitary matrices, in the
"scaling limit" as the size of the matrices goes to infinity. It
has been developed and used to model various phenomena in
mathematical physics. An exciting and relatively recent development in
the theory of zeta function is the realization that the imaginary
parts of the zeros, when viewed on the scale of their mean
separation, seem to have non-trivial statistics which are precisely
those appearing in Random Matrix Theory.
The goal of this seminar
is to explain what is known on the relation between zeros of the
Riemann zeta function and RMT. We will give tutorials on the basics of
RMT, the basic theory of the Riemann zeta function and related objects
(such as Dirichlet L-functions) and their applications in number
theory, and the state of the art in understanding the connections
between these two theories.
These are some of the lectures given during the course:
Here is some background material to look at before/during/after the school:
M.L. Mehta "Random Matrices", revised and enlarged second edition,
Academic press 1991
(the standard textbook on Random Matrix Theory).
H. Davenport "Multiplicative number theory". (The standard text on
analytic number theory).
N.M. Katz and P. Sarnak, "Random Matrices Frobenius Eigenvalues and
Monodromy", AMS Colloquium publications vol 45
(a research monograph).
- A. M. Odlyzko
The 10^20-th zero of the Riemann zeta function and 175 million of
its neighbors, (numerical evidence for several
conjectures about the zeros of zeta and the connection to RMT).
Zeev Rudnick's Homepage.