ETH :: D-MATH :: Wahrscheinlichkeitstheorie: Random Matrices and Number Theory

There exist important conjectures which relate the statistical behaviour of the
nontrivial zeros of the Riemann zeta function to the statistical behavior of the
eigenvalues of large random matrices, among which there is the celebrated GUE
hypothesis, also known as the Odlyzko-Montgomery law. In this series of
lectures, we shall introduce the basic techniques of random matrix theory (here
the unitary group endowed with the Haar measure) and some elementary facts about
the Riemann zeta function and primes, so that we can understand the GUE
conjecture and some more recent extensions of this conjecture. We then study the
characteristic polynomial of random unitary matrices and show how they have been
proved to be powerful in the moment conjecture for the Riemann zeta function
(the Keating-Snaith conjecture). These lectures will be mostly focused on the
probabilistic aspects and almost no number theory is required (in particular, we
will not deal with L-functions and elliptic curves).

Literatur

B. Conrey, A Guide to Random Matrix Theory For Number Theorists

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