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Residue Theorem, Regularization and Parity Theorem

Jia Li, Ce Xu

TL;DR

The work develops a contour-integral approach with residue calculus to establish explicit parity relations for (cyclotomic) multiple zeta values, introducing double shuffle regularization to produce two distinct parity formulas (shuffle and stuffle) that remain valid in the regularized setting, including the challenging case $k_r=1$. This method not only yields parity relations at arbitrary depth but also extends to cyclotomic and other variants of MZVs, creating a versatile framework for regularized parity phenomena. By leveraging a Flajolet–Salvy kernel and carefully analyzing residues, the authors derive finite-sum identities and regularization limits that underpin the parity formulas, with the regularization map $\rho$ bridging shuffle and stuffle sides. The results provide explicit parity decompositions and lay groundwork for broader applications in generalized zeta-valued objects, potentially impacting the study of regularized MZVs and their cyclotomic variants.

Abstract

In this paper, we employ contour integration and residue calculus to derive explicit parity formulas for (cyclotomic) multiple zeta values (MZVs). A key innovation lies in applying double shuffle regularization to the contour integrals, which leads to two distinct regularized parity formulas-one via shuffle and one via stuffle regularization. Notably, this demonstrates for the first time that the contour integral method can be extended to the regularized setting (including the case $k_r=1$), thereby overcoming a limitation of previous approaches. Our results not only provide explicit parity relations at arbitrary depths but also lay the groundwork for extending this technique to other variants of multiple zeta values.

Residue Theorem, Regularization and Parity Theorem

TL;DR

The work develops a contour-integral approach with residue calculus to establish explicit parity relations for (cyclotomic) multiple zeta values, introducing double shuffle regularization to produce two distinct parity formulas (shuffle and stuffle) that remain valid in the regularized setting, including the challenging case . This method not only yields parity relations at arbitrary depth but also extends to cyclotomic and other variants of MZVs, creating a versatile framework for regularized parity phenomena. By leveraging a Flajolet–Salvy kernel and carefully analyzing residues, the authors derive finite-sum identities and regularization limits that underpin the parity formulas, with the regularization map bridging shuffle and stuffle sides. The results provide explicit parity decompositions and lay groundwork for broader applications in generalized zeta-valued objects, potentially impacting the study of regularized MZVs and their cyclotomic variants.

Abstract

In this paper, we employ contour integration and residue calculus to derive explicit parity formulas for (cyclotomic) multiple zeta values (MZVs). A key innovation lies in applying double shuffle regularization to the contour integrals, which leads to two distinct regularized parity formulas-one via shuffle and one via stuffle regularization. Notably, this demonstrates for the first time that the contour integral method can be extended to the regularized setting (including the case ), thereby overcoming a limitation of previous approaches. Our results not only provide explicit parity relations at arbitrary depths but also lay the groundwork for extending this technique to other variants of multiple zeta values.
Paper Structure (5 sections, 25 theorems, 92 equations)

This paper contains 5 sections, 25 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Properties and Expansion
  3. Main Ideas
  4. Parity Conjecture and Regularization
  5. Parity Theorem for Cyclotomic Multiple Zeta Values

Key Result

Theorem 1.2

For all $r\in \mathbb{Z}_{>1}$ and ${\boldsymbol{\sl{n}}}=(n_1,\ldots,n_r)\in(\mathbb{Z}_{>0})^r$, the function is of depth at most $r-1$, meaning that it can be written as a $\mathbb{Q}$-linear combination of the functions where the indices ${{\boldsymbol{\sl{n}}}}^{(i)}\in \mathbb{N}^{d_i}$ have total depth $d_1+\cdots+d_s<d$ and preserve the weight $|{\boldsymbol{\sl{k}}}|+\sum_{i=1}^s |{\bol

Theorems & Definitions (49)

  • Conjecture 1.1: Borwein--Girgensohn Borwein-Girgensohn-1996, 1996
  • Theorem 1.2: Panzer Panzer2017, 2017
  • Definition 2.1
  • Example 2.1
  • Definition 2.2
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • ...and 39 more