The many faces of the stochastic zeta function

TL;DR

This paper introduces a new framework for studying the stochastic zeta function associated with the Sine$_eta$ process, providing explicit formulas, convergence rates, and connections to random matrix theory.

Contribution

It develops a comprehensive characterization of the stochastic zeta function, including explicit series, moment formulas, and convergence bounds, extending classical results to all beta values.

Findings

Explicit power series representation from Brownian motion

Upper bounds on convergence rate of characteristic polynomials

Validation of Borodin-Strahov moment formulas for all beta

Abstract

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