Mixed sums of infinite multiple zeta-star values with restricted indices
Abstract
We study arithmetic sums of infinite multiple zeta-star values subject to upper or lower bounds on their indices. Let η(𝒟q) be the image, under the zeta-star correspondence, of the admissible infinite index sequences with all indices at most q, and let η(𝒯p)̅ be the closure of the image of the sequences with all indices at least p. We prove that η(𝒟q)+η(𝒯p)̅ is an interval exactly when p≤ q; in that case it is [ζ⋆({q}∞)+1,∞). We also show that sums of two upper-bounded sets are half-lines, even with different bounds, whereas sums of two lower-bounded sets are never intervals. The proofs combine a uniform product estimate with natural gap constructions. Lateral thickness handles the diagonal case p=q≥3, while a non-strict local bridge--gap criterion treats p=q=2 directly. Compact truncations and explicit omitted intervals yield the remaining assertions.
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BibTeX
@misc{li2026mixed,
title = {{Mixed sums of infinite multiple zeta-star values with restricted indices}},
author = {Jiangtao Li and Siyu Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29100},
eprint = {2610.29100},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29100},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mixed sums of infinite multiple zeta-star values with restricted indices
AU - Jiangtao Li
AU - Siyu Yang
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29100
UR - https://arxiv.org/abs/2610.29100
AB - We study arithmetic sums of infinite multiple zeta-star values subject to upper or lower bounds on their indices. Let η(𝒟_q) be the image, under the zeta-star correspondence, of the admissible infinite index sequences with all indices at most q, and let η(𝒯_p)̅ be the closure of the image of the sequences with all indices at least p. We prove that η(𝒟_q)+η(𝒯_p)̅ is an interval exactly when p≤ q; in that case it is [ζ^⋆({q}^∞)+1,∞). We also show that sums of two upper-bounded sets are half-lines, even with different bounds, whereas sums of two lower-bounded sets are never intervals. The proofs combine a uniform product estimate with natural gap constructions. Lateral thickness handles the diagonal case p=q≥3, while a non-strict local bridge--gap criterion treats p=q=2 directly. Compact truncations and explicit omitted intervals yield the remaining assertions.
ER -