多重星多重对数的导出集
The derived set of multiple star polylogarithms
- Central South University（中南大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究单位圆上多重星多重对数的二维问题，证明其相关函数的单叶性、曲线分离性质，建立对应关系并应用于分圆多重ζ星值，得出其导出集为闭半平面Re w≥1/2。
AI中文摘要：
多重ζ星值的导出集是半直线[1, +∞)。本文研究单位圆上多重星多重对数对应的二维问题。我们首先证明每个移位多重星多重对数是完全单调序列的生成函数，因此具有归一化Hausdorff–Stieltjes表示。移位函数、未移位函数及其无穷深度极限均在半平面Re z<1上单叶。对于有限指标，代表密度关于逆字典序满足严格单调似然比序。结合Hausdorff–Stieltjes类的闭包性质与极限商论证，这表明沿上半圆，每条平移曲线的自变量严格递增而模严格递减；下半圆的结论通过共轭得到。同一思路可严格区分不同指标对应的曲线。随后，我们建立由单位圆上一点与无穷指标组成的自然对集合，与去掉点1的闭半平面Re w≥1/2之间的一一对应。在指标集的二元坐标下，该对应是同胚。作为应用，所有级别的移位分圆多重ζ星值的一个子类构成开半平面Re w>1/2的可数稠密子集。因此，该分圆族的导出集及所有高阶导出集均为闭半平面Re w≥1/2。
英文摘要：
The derived set of multiple zeta-star values is the half-line $[1,+\infty)$. In this paper we study the corresponding two-dimensional problem for multiple star polylogarithms on the unit circle. We first prove that every shifted multiple star polylogarithm is the generating function of a completely monotone sequence and hence admits a normalized Hausdorff--Stieltjes representation. Both the shifted and the unshifted functions, as well as their infinite-depth limits, are shown to be univalent on the half-plane $\mathrm{Re}\,z<1$. For finite indices, the representing densities satisfy a strict monotone likelihood-ratio order with respect to the reverse lexicographic order. Together with closure properties of the Hausdorff--Stieltjes class and a limiting quotient argument, this shows that, along the upper semicircle, the argument of each translated curve increases strictly while its modulus decreases strictly; the corresponding lower-semicircle statement follows by conjugation. The same idea gives strict separation of the curves attached to different indices. We then establish a one-to-one correspondence between a natural set of pairs consisting of a point of the unit circle and an infinite index, and the closed half-plane $\mathrm{Re}\,w\geq \frac12$ with the point $1$ removed. Under a binary coordinate on the index set, this correspondence is a homeomorphism. As an application, a subclass of shifted cyclotomic multiple zeta-star values of all levels form a countable dense subset of the open half-plane $\mathrm{Re}\,w>\frac12$. Consequently, the derived set, and every higher derived set, of this cyclotomic family is the closed half-plane $\mathrm{Re}\,w\geq\frac12$.
