The derived set of multiple star polylogarithms
Abstract
The derived set of multiple zeta-star values is the half-line [1,+∞). 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 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 Re w≥ 1/2 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 Re w>1/2. Consequently, the derived set, and every higher derived set, of this cyclotomic family is the closed half-plane Re w≥1/2.
Related
- Galois descents of certain multiple polylogarithms
- Graded Structures of v-adic Carlitz Multiple Polylogarithms and v-adic Multiple Zeta Values
- Multiple harmonic star sums involving central binomial coefficients as weights
- Mixed sums of infinite multiple zeta-star values with restricted indices
- Divided differences and complex variations of multiple zeta-star values
BibTeX
@misc{li2026derived,
title = {{The derived set of multiple star polylogarithms}},
author = {Jiangtao Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03653},
eprint = {2609.03653},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03653},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The derived set of multiple star polylogarithms
AU - Jiangtao Li
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03653
UR - https://arxiv.org/abs/2609.03653
AB - The derived set of multiple zeta-star values is the half-line [1,+∞). 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 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 Re w≥ 1/2 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 Re w>1/2. Consequently, the derived set, and every higher derived set, of this cyclotomic family is the closed half-plane Re w≥1/2.
ER -