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arXiv · 2604.03618

On $u$-Multiple Zeta Values in Positive Characteristic

Hung-Chun Tsui
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Abstract

In this paper, we introduce the concepts of the $u$-bracket, finite multiple harmonic $u$-series, and $u$-multiple zeta values via the Carlitz module. These objects serve as function field counterparts to the classical theory of $q$-analogs. We prove that the "limits" of finite multiple harmonic $u$-series at Carlitz torsion points yield Thakur's multiple zeta values and finite multiple zeta values over $\mathbb{F}_r(\theta)$ from analytic and algebraic perspectives, respectively. This can be regarded as a positive characteristic analog of the results by Bachmann, Takeyama, and Tasaka [BTT18]. Furthermore, we investigate the properties of $u$-multiple zeta values and their expansions, obtaining a family of explicit relations among Thakur's multiple zeta values at both positive and non-positive indices.

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Hung-Chun Tsui. 2026-04-04. On $u$-Multiple Zeta Values in Positive Characteristic. https://arxiv.org/abs/2604.03618

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