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

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.