2026/07/14 by Jonas Matuzas · 1 citation
Mathematics · #math.NT #math.RT
arxiv created 2026/08/03 · arxiv updated 2026/08/04
Let Φ be an irreducible crystallographic root system of rank r, with Coxeter number h, Weyl group W, Cartan matrix CΦ, and invariant degrees 2=d1≤⋯≤ dr=h. We prove that Au's normalized Witten zeta function has a simple pole at 2/h and evaluate its residue as \frac2(2π)r/2√(det CΦ)h|W|⋅\frac∏i<rΓ(1-di/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 A4 prediction.