Functional equations of axiomatic multiple Dirichlet series, Weyl groupoids, and quantum algebra

TL;DR

The paper develops an axiomatic framework for multivariable Dirichlet series whose coefficients are governed by five axioms, producing functional equations of Kubota-type and Dirichlet-type. These equations organize into a groupoid, identified with Weyl groupoids of arithmetic root systems, enabling a complete classification of finite-groupoid cases and a uniform construction of moments of L-functions. The authors connect analytic functional equations to geometry via traces of Frobenius on perverse sheaves and to quantum algebra through Nichols algebras, deriving cohomological identities and offering pathways to meromorphic continuation. The work unifies prior moments results, yields new moments, and suggests deep links between moments, automorphic objects, and categorical structures in quantum algebra. It also points toward further directions, including hybrid functional equations and number-field analogues, broadening the scope of function-field methods in analytic number theory.

Abstract

We prove functional equations for multiple Dirichlet series defined by a collection of five geometric axioms. We find functional equations of two types: one modeled on the functional equations of Dirichlet $L$-functions, and another modeled on the functional equations of Kubota $L$-series with Gauss sums as coefficients. These functional equations generate groupoid structures, which we relate to the Weyl groupoids of arithmetic root systems. From the known classification of arithmetic root systems, we obtain a complete classification of multiple Dirichlet series which can be used to compute moments of $L$-functions via established analytic techniques. Our classification includes all moments of $L$-functions which have appeared in the multiple Dirichlet series literature previously, alongside some new moments. Finally, we give applications of our functional equations to quantum algebra, specifically the cohomology of Nichols algebras.

Figures (1)

• Figure 1: Weyl groupoid of functional equations.

Theorems & Definitions (123)

• Theorem 1.1
• Theorem \ref{TheoremMultiFE}
• Theorem \ref{dirichlet-multivariable-global}
• Theorem \ref{rational-fn}
• Theorem \ref{uniquely-determined}
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof