Physicists developed Random Matrix Theory (RMT) in the 1950s to explain the energy levels of heavy nuclei. A fortuitous meeting over tea at the Institute in the 1970s revealed that similar answers are found for zeros of L-functions, and since then RMT has been used to model their behavior. The distribution of these zeros is intimately connected to many problems in number theory, from how rapidly the number of primes less than X grows to the class number problem to the bias of primes to be congruent to 3 mod 4 and not 1 mod 4. The Katz and Sarnak density conjectures state that in the limit the behavior of scaled zeros in a family of L-functions agrees with the scaling limits of eigenvalues of a classical compact group.

We report on recent progress on the n-level densities of low-lying zeros of GL(2) L-functions. We derive an alternate formula for the Katz-Sarnak determinant expansions for test functions with large support that facilitates comparisons between number theory and random matrix theory in orthogonal, symplectic, and unitary settings. Using combinatorics, generating functions, and analysis, we prove these formulas hold and increase the region where number theory and random matrix theory can be shown to agree for holomorphic cuspidal newforms. If time permits we will discuss extensions to other GL(2) families. This is joint work with Nicholas Triantafillou (Michigan) and Levent Alpoge (Harvard).