On $u$-Multiple Zeta Values in Positive Characteristic
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917383880310784 |
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| author | Tsui, Hung-Chun |
| author_facet | Tsui, Hung-Chun |
| contents | 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 $\mathbb{F}_r(θ)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03618 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On $u$-Multiple Zeta Values in Positive Characteristic Tsui, Hung-Chun Number Theory Primary 11M32, Secondary 11R58 11R59 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 $\mathbb{F}_r(θ)$ 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. |
| title | On $u$-Multiple Zeta Values in Positive Characteristic |
| topic | Number Theory Primary 11M32, Secondary 11R58 11R59 |
| url | https://arxiv.org/abs/2604.03618 |