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On $u$-Multiple Zeta Values in Positive Characteristic

Hung-Chun Tsui

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 $\mathbb{F}_r(θ)$ 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.

On $u$-Multiple Zeta Values in Positive Characteristic

Abstract

In this paper, we introduce the concepts of the -bracket, finite multiple harmonic -series, and -multiple zeta values via the Carlitz module. These objects serve as function field counterparts to the classical theory of -analogs. We prove that the "limits" of finite multiple harmonic -series at Carlitz torsion points yield Thakur's multiple zeta values and finite multiple zeta values over 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 -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.

Paper Structure

This paper contains 17 sections, 38 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. Classical q-Analog Theory
  3. Classical Theory of Multiple Zeta Values
  4. Preliminaries on Function Field Arithmetic
  5. Main Results and Organization of the Paper
  6. u-Bracket and u-Carlitz Factorial
  7. u-Bracket and u-Carlitz Factorial
  8. Estimates for the u-Bracket
  9. Finite Multiple Harmonic u-Series
  10. Finite Multiple Harmonic u-Series
  11. r-Shuffle Relations
  12. Euler--Carlitz-type Formula for H<d at Carlitz Torsion Points
  13. "Limits" of H<d at Carlitz Torsion Points
  14. u-Multiple Zeta Values
  15. u-Multiple Zeta Values on Drinfeld Upper-Half Plane
  16. ...and 2 more sections

Key Result

Theorem 1.2

For any index $\mathfrak{s}\in\mathbf{I}$, we have $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (85)

  • Conjecture 1.1: Kaneko--Zagier Conjecture
  • Theorem 1.2: BTT18
  • Theorem 1.3: BTT18
  • Definition 1.4
  • Theorem 1.5: Shi2018
  • Theorem 1.6: restated as Theorem \ref{['thm:limit of u']}
  • Theorem 1.7: restated as Theorem \ref{['thm:u to finite MZV']}
  • Theorem 1.8: restated as Propositions \ref{['prop:expansion of umzv']} and \ref{['prop:gamma mzv']}
  • Theorem 1.9: restated as Theorem \ref{['thm:D_N_derivation_rigorous']}
  • Definition 2.1
  • ...and 75 more