Distributions of sums of the divisor function over function fields

The goal of this talk is to discuss the variance of sums of the divisor function leading to certain random matrix distributions. While the knowledge of these problems is quite limited over the natural numbers, much more is known over function fields. We will start by introducing the basics of zeta functions and $L$-functions over function fields. We will then discuss the work of Keating, Rodgers, Roditty-Gershon, and Rudnick on the sums over arithmetic progressions, leading to distributions over unitary matrices by the Katz and Sarnak philosophy and a general conjecture over the natural numbers. Finally, we will present some recent work (in collaboration with Kuperberg) on sums over squares modulo a prime leading to symplectic distributions.