{"paper":{"title":"Critical probabilistic characteristics of the Cram\\'er model for primes and arithmetical properties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Michel Weber","submitted_at":"2021-05-23T21:13:10Z","abstract_excerpt":"This work is a probabilistic study of the 'primes' of the Cram\\'er model. We prove that there exists a set of integers $\\mathcal S$ of density 1 such that \\begin{equation}\\liminf_{ \\mathcal S\\ni n\\to\\infty} (\\log n)\\mathbb{P} \\{S_n\\ \\hbox{prime} \\} \\ge \\frac{1}{\\sqrt{2\\pi e}\\, }, \\end{equation} and that for $b>\\frac12$, the formula \\begin{equation} \\mathbb{P} \\{S_n\\ \\text{prime}\\, \\} \\, =\\, \\frac{ (1+ o( 1) )}{ \\sqrt{2\\pi B_n } } \\int_{m_n-\\sqrt{ 2bB_n\\log n}}^{m_n+\\sqrt{ 2bB_n\\log n}} \\, e^{-\\frac{(t - m_n)^2}{ 2 B_n } }\\, {\\rm d}\\pi(t), \\end{equation} in which $m_n=\\mathbb{E} S_n,B_n={\\rm Va"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.11020","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/2105.11020/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"}