Random matrix models for Gram's law

Gram's Law refers to the empirical observation that the zeros of the Riemann zeta function typically alternate with certain prescribed points, called Gram points. Although this pattern does not hold true for each and every zero, numerical results suggest that, as the height up the critical line increases, the proportion of zeros that obey Gram's Law converges to a finite, non-zero limit. It is also well-known that the eigenvalues of random unitary matrices provide a good statistical model for the distribution of zeros of the zeta function, so one could try to determine the value of this limit by analyzing an analogous model for Gram's Law in the framework of Random Matrix Theory. In this thesis, we will review an existing model based on random unitary matrices, for which the limit can be computed analytically, but has the wrong rate of convergence. We will then present an alternative model that uses random special unitary matrices, which gives the correct convergence rate, and discuss the large-N limit of this model. We shall conclude that at very large heights up the critical line, the local distribution of the zeta zeros is the same with respect to any sequence of points that are spaced like the Gram points. For the purpose of this thesis, we will assume throughout that all Gram points are different from zeta zeros, although this is not a proven fact.