The many faces of the stochastic zeta function

We introduce a framework to study the random entire function $\zeta_\beta$ whose zeros are given by Sine$_\beta$ process, bulk limit of beta ensembles. present several equivalent characterizations, including an explicit power series representation built from Brownian motion. related distributions using stochastic differential equations. Our is uniform characteristic polynomials in circular ensemble; we give upper bounds on rate convergence. Most our results new even for classical values $\beta$. provide moment formulas $\zeta$ and its variants, show that Borodin-Strahov hold all $\beta$ both uniqueness theorem Cartwright class, deduce some product identities between conjugate The proofs rely structure operator express terms regularized determinant.