An overview and supplements to the theory of functional relations for zeta-functions of root systems

TL;DR

This paper reviews the theory of functional relations for zeta-functions of root systems and introduces new results for types B_r, D_r, A_3, and C_2 using generating functions and elementary methods.

Contribution

It provides new functional relations for specific root system zeta-functions using two distinct methods, expanding the theoretical understanding.

Findings

New functional relations for B_r, D_r, A_3, C_2 zeta-functions

Use of generating functions based on Weyl group symmetry

Elementary partial fraction method for C_2

Abstract

We give an overview of the theory of functional relations for zeta-functions of root systems, and show some new results on functional relations involving zeta-functions of root systems of types $B_r$, $D_r$, $A_3$ and $C_2$. To show those new results, we use two different methods. The first method, for $B_r$, $D_r$, $A_3$, is via generating functions, which is based on the symmetry with respect to Weyl groups, or more generally, on our theory of lattice sums of certain hyperplane arrangements. The second method for $C_2$ is more elementary, using partial fraction decompositions.