In this talk we present a new approximate functional equation for the Hardy Z function with exponentially decaying error. This formula makes it possible to define a variation space for Z(t) consisting of a family of functions Z_N(t,a) with parameters a = (a1, …, aN) on a fixed window [2n, 2N + 2]. In this space the original Hardy function Z(t) is extremely well approximated in the window by the special choice a = (1, …, 1). Within this variation space we single out a natural domain RH_N in which all functions Z_N(t,a) have only real zeros in the window. We show that RH_N has striking connections with random matrix theory. In particular a Brownian motion with Skorokhod type reflection at the boundary of RH_N induces a Dyson Brownian motion in the \beta=2 case, and modern universality results then yield the expected GUE local statistics for elements of RH_N. No prior knowledge in number theory would be required.
