Vanishing of Witten zeta function at negative integers

TL;DR

This paper proves that the Witten zeta function associated with root systems vanishes to high order at negative even integers, confirming a conjecture for many compact Lie groups and linking the leading coefficient to Riemann zeta values.

Contribution

It establishes the high-order vanishing of the Witten zeta function at negative even integers and describes the leading coefficient in terms of Riemann zeta values, settling a conjecture for a broad class of groups.

Findings

High-order vanishing at negative even integers.

Integral representation involving Hurwitz zeta function.

Leading coefficient expressed via Riemann zeta values.

Abstract

We prove Witten zeta function of a root system $\Phi$ has high-order vanishing at negative even integers, using an integral representation involving the Hurwitz zeta function. This settles a conjecture of Kurokawa and Ochiai for a large class of compact Lie groups. We also provide a qualitative description of the corresponding leading coefficient in terms of Riemann zeta values, in which the highest root of $\Phi$ makes a natural appearance.