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Symmetry Results for Cyclotomic Multiple Hurwitz Zeta Values via Contour Integrals

Ce Xu

TL;DR

This work extends symmetry results from cyclotomic regularized t-values to cyclotomic multiple Hurwitz zeta values with multiple variables and parameters by employing a purely analytic contour-integral approach. It provides explicit, regularization-free symmetry formulas for both cyclotomic multiple zeta values and cyclotomic Hurwitz zeta values, and derives concise mod-product forms by omitting lower-depth product terms. The two central theorems are complemented by corollaries that recover cyclotomic t-values and verify conjectures such as Rui’s in appropriate specializations. The method relies on residue calculations with carefully constructed kernel functions and expansions, offering a robust framework for exploring symmetry phenomena in related zeta-value variants and suggesting potential extensions to other zeta-function families.

Abstract

This paper provides a systematic study of symmetry properties for cyclotomic multiple Hurwitz zeta values with multiple variables and parameters by applying the methods of contour integration and the residue theorem. The main contributions are the derivation of explicit symmetry formulas for cyclotomic multiple (Hurwitz) zeta values, which are obtained directly through analytic residue calculations, without reliance on algebraic regularization. As a concrete application, we deduce analogous symmetry theorems for cyclotomic multiple zeta values and cyclotomic multiple $t$-values. The results extend and complement recent symmetry investigations by Charlton and Hoffman, offering completely explicit and regularization-free formulas in the convergent setting. Moreover, the results of this paper can be used to prove the symmetry conjecture for cyclotomic multiple Hurwitz zeta values with multiple variables and a single parameter. Furthermore, several illustrative corollaries and examples are included, and an open problem concerning possible extensions to other variants of multiple zeta values is posed at the conclusion.

Symmetry Results for Cyclotomic Multiple Hurwitz Zeta Values via Contour Integrals

TL;DR

This work extends symmetry results from cyclotomic regularized t-values to cyclotomic multiple Hurwitz zeta values with multiple variables and parameters by employing a purely analytic contour-integral approach. It provides explicit, regularization-free symmetry formulas for both cyclotomic multiple zeta values and cyclotomic Hurwitz zeta values, and derives concise mod-product forms by omitting lower-depth product terms. The two central theorems are complemented by corollaries that recover cyclotomic t-values and verify conjectures such as Rui’s in appropriate specializations. The method relies on residue calculations with carefully constructed kernel functions and expansions, offering a robust framework for exploring symmetry phenomena in related zeta-value variants and suggesting potential extensions to other zeta-function families.

Abstract

This paper provides a systematic study of symmetry properties for cyclotomic multiple Hurwitz zeta values with multiple variables and parameters by applying the methods of contour integration and the residue theorem. The main contributions are the derivation of explicit symmetry formulas for cyclotomic multiple (Hurwitz) zeta values, which are obtained directly through analytic residue calculations, without reliance on algebraic regularization. As a concrete application, we deduce analogous symmetry theorems for cyclotomic multiple zeta values and cyclotomic multiple -values. The results extend and complement recent symmetry investigations by Charlton and Hoffman, offering completely explicit and regularization-free formulas in the convergent setting. Moreover, the results of this paper can be used to prove the symmetry conjecture for cyclotomic multiple Hurwitz zeta values with multiple variables and a single parameter. Furthermore, several illustrative corollaries and examples are included, and an open problem concerning possible extensions to other variants of multiple zeta values is posed at the conclusion.
Paper Structure (5 sections, 18 theorems, 73 equations)

This paper contains 5 sections, 18 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Main Theorems and Corollaries
  3. Lemmas and Expansions
  4. Symmetry Results of Cyclotomic Multiple Zeta Values
  5. Symmetry Results of Cyclotomic Multiple Hurwitz Zeta Values

Key Result

Theorem 2.1

Let $k_0=q$ and $x_0=x$. For $(k_0,k_1,\ldots,k_r)\in \mathbb{N}^{r+1}$ and $x_0,x_1,\ldots,x_r$ are roots of unity with $(k_r,x_r)\neq(1,1)$ and $(k_0,x_0)\neq (1,1)$, we have where $\widetilde{{\boldsymbol{\sl{x}}}_{0,r}}:=x_0x_1\cdots x_r,\ \widetilde{{\boldsymbol{\sl{x}}}^{-1}_{0,r}}:=(x_0x_1\cdots x_r)^{-1}$, and $\mathop{\mathrm{Li}}\nolimits_0(x)+\mathop{\mathrm{Li}}\nolimits_0(x^{-1}):=-1

Theorems & Definitions (26)

  • Conjecture 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Remark 2.7
  • Theorem 2.8
  • proof
  • ...and 16 more