The derived set of multiple star polylogarithms

Abstract

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 $\mathrm{Re}\,w\geq\frac12$.