Random Matrices and L-Functions 2.

Brian Conrey

(Joint mini-series with Nina Snaith)

Abstract: That a connection exists between random matrix theory and number theory has been known ever since the 70s when H.L.Montgomery and F.J.Dyson discovered over a cup of tea that the statistics of the zeros of the Riemann zeta function calculated by the former agree with the eigenvalue distribution of random matrices examined by the latter. This has led to random matrix theory being applied in recent years to surprisingly diverse areas of number theory, such as mean values of the Riemann zeta function and its derivatives, ranks of elliptic curves and even to curious arithmetic problems with seemingly no connection to random matrices at all. Random matrix theory continues to surprise us with the depth and diversity of its applications in number theory and we hope to demonstrate this by the examples presented in the lecture.