{"paper":{"title":"The De Bruijn-Newman constant is non-negative","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Brad Rodgers, Terence Tao","submitted_at":"2018-01-18T02:47:14Z","abstract_excerpt":"For each $t \\in {\\bf R}$, define the entire function $$ H_t(x) := \\int_0^\\infty e^{tu^2} \\Phi(u) \\cos(xu)\\ du$$ where $\\Phi$ is the super-exponentially decaying function $$ \\Phi(u) := \\sum_{n=1}^\\infty (2\\pi^2 n^4 e^{9u} - 3\\pi n^2 e^{5u} ) \\exp(-\\pi n^2 e^{4u} ).$$ Newman showed that there exists a finite constant $\\Lambda$ (the \\emph{de Bruijn-Newman constant}) such that the zeroes of $H_t$ are all real precisely when $t \\geq \\Lambda$. The Riemann hypothesis is the equivalent to the assertion $\\Lambda \\leq 0$, and Newman conjectured the complementary bound $\\Lambda \\geq 0$.\n  In this paper w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.05914","kind":"arxiv","version":5},"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/1801.05914/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"}