A Riemann-type duality of shuffle Hopf algebras related to multiple zeta values

TL;DR

This paper establishes a duality between two Hopf algebras related to multiple zeta values, providing an algebraic interpretation of the functional equation of the Riemann zeta function.

Contribution

It introduces a duality between shuffle Hopf algebras for positive and nonpositive arguments of MZVs, unifying them in a single algebraic framework.

Findings

Identifies a differential Hopf algebra structure for nonpositive MZV arguments.

Establishes a graded duality between the algebraic structures for positive and nonpositive MZVs.

Provides an explicit isomorphism modeling the Riemann zeta functional equation.

Abstract

This paper offers a Hopf algebraic interpretation of a functional equation of multiple zeta functions, motivated by the classical symmetry of the Riemann zeta function. Starting from the extended shuffle algebra that encodes multiple zeta values (MZVs) at integer arguments, we show that its subalgebra corresponding to nonpositive arguments carries a natural differential Hopf algebra structure. This Hopf algebra is in graded linear duality with the shuffle Hopf algebra associated to MZVs at positive arguments. The resulting duality, realized through an explicit isomorphism, provides an algebraic analog of the functional equation relating $\zeta(s)$ with $\zeta(1-s)$ of the Riemann zeta function and unifies the positive and nonpositive sectors of multiple zeta functions within a common Hopf algebraic framework.