Double shuffle relations for refined symmetric multiple zeta values

TL;DR

This paper introduces refined symmetric multiple zeta values (RSMZVs), lifts of symmetric multiple zeta values that satisfy refined double shuffle and duality relations, connecting them to existing variants like the ξ-values.

Contribution

The paper constructs RSMZVs as iterated integrals, establishing their double shuffle and duality relations, and shows their equivalence to known ξ-values.

Findings

RSMZVs satisfy refined double shuffle relations.

RSMZVs obey duality relations.

RSMZVs coincide with ξ-values of Bachmann-Takeyama-Tasaka.

Abstract

Symmetric multiple zeta values (SMZVs) are elements in the ring of all multiple zeta values modulo the ideal generated by $\zeta(2)$ introduced by Kaneko-Zagier as counterparts of finite multiple zeta values. It is known that symmetric multiple zeta values satisfy double shuffle relations and duality relations. In this paper, we construct certain lifts of SMZVs which live in the ring generated by all multiple zeta values and $2\pi i$ as certain iterated integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty\}$ along a certain closed path. We call this lifted values as refined symmetric multiple zeta values (RSMZVs). We show double shuffle relations and duality relations for RSMZVs. These relations are refinements of the double shuffle relations and the duality relations of SMZVs. Furthermore we compare RSMZVs to other variants of lifts of SMZVs. Especially, we prove that RSMZVs coincide with Bachmann-Takeyama-Tasaka's $\xi$-values.