Some remarks on the order structures of multi-polylogarithms

TL;DR

This paper studies a variant of multi-polylogarithm functions, exploring their integral representations, order structures, and conditions for their values to be dense, contributing to the understanding of their algebraic and topological properties.

Contribution

It introduces a new class of multi-polylogarithm functions with unique integral representations and characterizes when their values form a dense set.

Findings

Multi-polylogarithms have two integral representations.

Their order structure resembles that of multiple zeta star values.

A necessary and sufficient condition for density of their values is provided.

Abstract

We consider multi-polylogarithm functions which are slightly different from the ordinary ones. These functions have two integral representations and an order structure similar to those of multiple zeta star values. We also give a necessary and sufficient condition for the set of values of our multi-polylogarithm functions to be a dense set.