Residue Theorem, Regularization and Parity Theorem

TL;DR

This paper develops a novel contour integral approach with regularization techniques to derive explicit parity formulas for multiple zeta values, extending the method's applicability to previously inaccessible cases.

Contribution

It introduces a new regularized contour integral method for multiple zeta values, enabling explicit parity formulas at arbitrary depths, including the challenging case where $k_r=1$.

Findings

Derived explicit parity formulas for MZVs using contour integrals.

Extended the contour integral method to regularized cases, including $k_r=1$.

Established groundwork for applying this technique to other MZV 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.