The topology of the set of multiple zeta-star values
Jiangtao Li
TL;DR
This work establishes a topological and combinatorial framework for the set of multiple zeta-star values $\zeta^{\star}(k_1,\dots,k_r)$. It provides a universal multiple-integral representation and builds an order that makes $\zeta^{\star}$ strictly monotone with respect to index tuples, then constructs a bijection $\eta:\mathcal{T}\to(1,\infty)$ preserving this order, yielding a dense image of $\mathcal{Z}^{\star}$ in $(1,\infty)$ and a full description of accumulation points via a second map $\xi:\mathcal{A}\to(1,\infty)$. The paper further analyzes the measure-theoretic and fractal properties of the image sets, deriving Hausdorff-dimension formulas for Cantor-like subsets and proving that almost all accumulation points come from unbounded index sequences. By computing explicit limits for natural families $\zeta^{\star}(\cdots, \{p\}^m)$ and identifying connections to Riemann zeta values through alternative integral representations, it shows that all values are non-integers and discusses implications for irrationality and transcendence of these numbers. Overall, the results illuminate the topology and distribution of $\zeta^{\star}$-values and connect them to fractal geometry and classical zeta theory.
Abstract
We provide a multiple integral representation for each multiple zeta-star value, and utilize these representations to establish a natural order structure on the set of such values. This order structure allows for a one-to-one correspondence between a subset of the infinite sequences of natural numbers and the half line $(1,+\infty)$. Some basic properties of this correspondence are discussed. We also calculate the Hausdorff dimensions for the images of some subsets of the infinite sequences under this correspondence. As a result of this correspondence, we are able to determine the limits for a number of natural multiple integrals. Our analysis also reveals that the set of multiple zeta-star values is dense within the $(1,+\infty)$ domain, and that each value is non-integer in nature.