We review some of the connections between Random Matrix Theory and Number Theory. These include modelling the value distributions of the Riemann zeta-function and other L-functions, and the statistical distribution of their zeros. 1.

Random-matrix theory gained attention during the 1950s due to work by Eugene Wigner in mathematical physics. Specifically, Wigner wished to describe the general properties of the energy levels (or of their spacings) of highly excited states of heavy nuclei as measured in nuclear reactions (Wigner, 1957). Such a complex nuclear system is represented by an Hermitian operator H (called the Hamilto...

3 Universality 4 3.1 Macroscopic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1.1 Wigner’s semicircle law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Microscopic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2.1 Bulk universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...