Au’s gamma-product conjecture for Witten zeta functions

Au’s conjecture predicts that the leading residue of each normalized Witten zeta function has an algebraic factor times products of gamma values at rational arguments. The 2026 theorem establishes this uniformly for all irreducible crystallographic root systems.

Known results

• Au computed the leading residues in ranks two and three.
• Au recorded numerical evidence for type $A_4$, including the predicted value $R_{A_4}=\frac{\sqrt{50-22\sqrt{5}}}{240\pi}\Gamma(1/5)^5$.
• The 2024 rank-two-and-three paper stated the conjecture and reported supporting numerical evidence, but did not establish it in all ranks.

July 2026 universal proof

Theorem 2.1 of “A universal leading-residue formula for Witten zeta functions” proves that $\xi_{\Phi}(s)$ has a simple pole at $s=2/h$ and gives its residue explicitly as a quotient involving $\Gamma(1-d_i/h)$ and $\Gamma(1-1/h)$. This proves Au’s conjecture in every allowed type and confirms the $A_4$ prediction; the ordinary-residue formula follows as well.

Current status (as of July 2026): Au’s gamma-product conjecture is settled for all irreducible crystallographic root systems; no case remains open.