On single-variable Witten zeta functions of rank two and three

TL;DR

This paper investigates the properties of Witten zeta functions of rank two and three, revealing new features through a novel Mellin transform approach, including pole structures, residues, special values, and conjectures about derivatives.

Contribution

It introduces a new integration kernel for Mellin transforms, uncovering previously unknown properties of these zeta functions and proposing a conjecture linking derivatives to root system data.

Findings

New pole and residue structures identified

Special values of the zeta functions characterized

A conjecture relating derivatives at zero to root systems proposed

Abstract

By introducing a novel integration kernel for Mellin transform, we uncover many previously unknown and intriguing properties of the Witten zeta functions of rank two and three. Detailed results concerning their pole locations, residues, and special values are obtained. We propose a non-trivial conjecture regarding their derivatives at the origin, which seems to encode deep information about the root system. We also discuss their behavior at negative integers, highlighting a connection with Eisenstein series and a $p$-adic observation.