The topology of the set of multiple zeta-star values

TL;DR

This paper introduces a new order structure on multiple zeta-star values using integral representations, revealing their density, non-integer nature, and connections to infinite sequences and Hausdorff dimensions.

Contribution

It establishes a natural order on multiple zeta-star values and explores their topological and measure-theoretic properties, including a correspondence with the half-line and Hausdorff dimensions.

Findings

Set of multiple zeta-star values is dense in (1, +∞).

Established a one-to-one correspondence with infinite sequences.

Determined limits of certain multiple integrals.

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.