{"paper":{"title":"Derivative sums of balanced gamma quotients and multiple zeta values: four conjectures of Zhi-Wei Sun","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Shivam Nalin Patel","submitted_at":"2026-07-14T18:27:57Z","abstract_excerpt":"We introduce a uniform reduction for derivative sums of balanced gamma quotients. For exponent data $(a_i,e_i)$ satisfying $\\sum_i e_i a_i=0$, the entire translation-dependent gamma prefactor is governed by the characteristic power sums $\\chi_m=\\sum_i e_i a_i^m$ through the identity $\\log C(u)=\\sum_{m\\geq2}(-1)^m\\chi_m\\zeta(m)u^m/m$. This separates the analytic gamma contribution from a hypergeometric coefficient-extraction problem and explains, in a single framework, the inverse-$4$ and inverse-$27$ families occurring in four conjectures of Zhi-Wei Sun. For the branch $\\chi_m=2-2^m$, diagonal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.27229","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/2607.27229/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"}