Probabilistic Models for Gram's Law.

Gram's law refers to the observation that the zeros of the Riemann zeta function typically alternate with certain prescribed points, known as Gram points. Although this pattern doesn't hold for 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 non-trivial limit. It is well-known that the eigenvalues of random unitary matrices provide a good statistical model for the distribution 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 paper, 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-height limit of this model.