Summer School on The Riemann Zeta Function and Random Matrix Theory

Oberwolfach, October 15 - 21, 2000

Organizers:

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.

Course materials:

These are some of the lectures given during the course:

Zeta functions in arithmetic and their spectral statistics by Zeev Rudnick.
(Outdated as far as RMT goes).
Weyls' integration formula on U(N), by Angela Pasquale.
Notes on Random Matrix Theory, by Nina Snaith.
Uniform distribution, by Christoph Baxa.
Level spacing statistics in number theory and mathematical physics, by Par Kurlberg.
Artin's primitive root conjecture - a survey, by Pieter Moree.
Value distribution of L(1,x), by Manfred Peter.
Central limit theorems and large deviations for characteristic polynomials of random unitary matrices and the Riemann zeta function, by Chris Hughes.

Background reading:

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).

participants.

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