{"paper":{"title":"A universal leading-residue formula for Witten zeta functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.RT","authors_text":"Jonas Matuzas","submitted_at":"2026-07-14T12:58:23Z","abstract_excerpt":"Let $\\Phi$ be an irreducible crystallographic root system of rank $r$, with Coxeter number $h$, Weyl group $W$, Cartan matrix $C_\\Phi$, and invariant degrees $2=d_1\\leq\\cdots\\leq d_r=h$. We prove that Au's normalized Witten zeta function $\\xi_\\Phi(s)$ has a simple pole at $s=2/h$, with residue $\\mathop{\\rm Res}_{s=2/h}\\xi_\\Phi(s)=\\frac{2(2\\pi)^{r/2}\\sqrt{\\det C_\\Phi}}{h|W|}\\frac{\\prod_{i=1}^{r-1}\\Gamma(1-d_i/h)}{\\Gamma(1-1/h)^r}$. The proof identifies the leading lattice coefficient with a convergent spherical Coxeter-discriminant integral at the critical exponent and evaluates this integral u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.12728","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.12728/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}