{"paper":{"title":"A new hyperbolicity wedge and a joint semicircle limit for Jensen polynomials of Riemann's $\\xi$-function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jonathan Holland","submitted_at":"2026-08-09T12:58:52Z","abstract_excerpt":"Let \\[\n  \\xi\\!\\left(\\frac12+z\\right)\n  =\\sum_{n\\geq 0}\\frac{\\gamma(n)}{n!}z^{2n},\n  \\qquad\n  J^{d,n}(X)\n  =\\sum_{j=0}^{d}\\binom dj\\gamma(n+j)X^j . \\] The Riemann hypothesis is equivalent to the hyperbolicity of $J^{d,n}$ for every $d,n\\geq0$. We prove that there is an absolute constant $K>0$ such that \\[\n  n^3\\log^2(n+2)\\geq Kd^5\n  \\quad\\Longrightarrow\\quad\n  J^{d,n}\\ \\text{is hyperbolic}. \\] Along every sequence with $n,d\\to\\infty$ in this region, the empirical measure of the naturally centered and scaled zeros also converges to Wigner's semicircle law. This gives a simultaneous degree--deriv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.08682","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/2608.08682/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"}