Double shuffle and naive duality for multiple Dirichlet series

Abstract

Multiple zeta values admit both iterated-sum and iterated-integral representations, giving rise to the stuffle, shuffle and duality relations. Although these relations are satisfied by many families of complex numbers, we show that multiple zeta values are essentially uniquely determined by the double shuffle relations within a natural class of geometric multiple Dirichlet series. We then study geometric multiple Dirichlet series satisfying the shuffle relations and a duality relation. Under suitable analytic assumptions, we show that such dualities arise from differentiable involutions. We also use this framework to derive linear functional relations among multiple polylogarithms. Finally, we consider a more naive notion of multiple Dirichlet series and discuss the restrictions imposed by the stuffle relations, together with a conjecture on the uniqueness of multiple zeta values in this setting.