A universal leading-residue formula for Witten zeta functions

Abstract

Let $Φ$ be an irreducible crystallographic root system of rank $r$, with Coxeter number $h$, Weyl group $W$, Cartan matrix $C_Φ$, and invariant degrees $2=d_1\leq\cdots\leq d_r=h$. We prove that Au's normalized Witten zeta function has a simple pole at $2/h$ and evaluate its residue as $\frac{2(2π)^{r/2}\sqrt{\det C_Φ}}{h|W|}\cdot\frac{\prod_{i<r}Γ(1-d_i/h)}{Γ(1-1/h)^r}$. The central step evaluates the critical chamber integral in gamma values from the boundary pole of the Macdonald-Mehta-Opdam identity; two preparatory sections pass from the dominant-weight lattice to that convergent integral. This proves Au's conjecture on algebraic multiples of products of gamma values at rational arguments, including his $A_4$ prediction.

Disclosure

“a integral; finite Coxeter group; root system; lattice-point asymptotics. Generative-AI disclosure and author responsibility. This work was produced using OpenAI’s ChatGPT 5.6 Pro. The author directed and audited the work throughout. Jonas Matuzas takes full responsibility for the mathematics and the final text. 1 Introd”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T18:55:14.641513Z

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