论文标题
随机Zeta功能的许多面
The many faces of the stochastic zeta function
论文作者
论文摘要
我们引入了一个框架，以研究随机整个函数$ζ_β$，其零是由正弦$_β$过程给出的，这是beta集成的批量极限。我们提出了几种等效的特征，包括由布朗运动构建的明确的功率序列表示。 我们使用随机微分方程研究相关的分布。我们的功能是圆形β集合中特性多项式的均匀极限。我们给出了收敛速度的上限。即使对于$β$的经典值，我们的大多数结果都是新的。 我们为$ζ$及其变体提供明确的力矩公式，并表明borodin-strahov矩配方在极限和圆形beta集合中均适用于所有$β$。我们在Cartwright类中显示了$ζ$的唯一定理，并在$β$的共轭值之间推断出一些产品标识。这些证明依赖于正弦$_β$运算符的结构来表达$ζ$，以正规化的决定因素表示。
We introduce a framework to study the random entire function $ζ_β$ whose zeros are given by the Sine$_β$ 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 $β$. We provide explicit moment formulas for $ζ$ and its variants, and we show that the Borodin-Strahov moment formulas hold for all $β$ both in the limit and for circular beta ensembles. We show a uniqueness theorem for $ζ$ in the Cartwright class, and deduce some product identities between conjugate values of $β$. The proofs rely on the structure of the Sine$_β$ operator to express $ζ$ in terms of a regularized determinant.
