The many faces of the stochastic zeta function (Q2088130)

An article is devoted to the study of the stochastic zeta function \(\zeta_\beta\). The authors start with a deep review of results obtained by many mathematicians, which gave the basis for this study of the random entire function \(\zeta_\beta\). They present several equivalent characterizations of the function \(\zeta_\beta\), particularly, an explicit power series representation built from Brownian motion, and, using stochastic differential equations, studied related distributions. Since the function \(\zeta_\beta\) is a uniform limit of characteristic polynomials in the circular beta ensemble, upper bounds on the rate of convergence are obtained by authors too. As well they give explicit moment formulas for \(\zeta\) and its variants and prove the Borodin-Strahov moment formulas for all \(\beta\) (both in the limit and for circular beta ensembles). One more result, obtained by the researchers, is a uniqueness theorem for \(\zeta\) in the Cartwright class.