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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2105.11020","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-05-23T21:13:10Z","cross_cats_sorted":[],"title_canon_sha256":"db031d063cde745965a9a605d2d289c64214aa4ab1d36e40be0ecd0fb00a6eee","abstract_canon_sha256":"68230e6e49b0c4fc2da6fa909f2507c6275d10a6bdfcd732037ab9a19b1b4386"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:03:38.415467Z","signature_b64":"X3U3XKy8OFtB9OgCHxU+6akDYDSUAJrs3eBiJN0HD5LJvaFry1yhwt+beBBusRNnuRsmyYnN91mjJHwFcCNcAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cd63f9dfe3ba1e1a3c78680bf1ead32d8a64fd52e97aa8d98bb869b97f8dd729","last_reissued_at":"2026-05-22T01:03:38.414280Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:03:38.414280Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2105.11020","created_at":"2026-05-22T01:03:38.414405+00:00"},{"alias_kind":"arxiv_version","alias_value":"2105.11020v1","created_at":"2026-05-22T01:03:38.414405+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2105.11020","created_at":"2026-05-22T01:03:38.414405+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZVR7TX7DXIPB","created_at":"2026-05-22T01:03:38.414405+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZVR7TX7DXIPBUPDY","created_at":"2026-05-22T01:03:38.414405+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZVR7TX7D","created_at":"2026-05-22T01:03:38.414405+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZVR7TX7DXIPBUPDYNAF7D2WTFW","json":"https://pith.science/pith/ZVR7TX7DXIPBUPDYNAF7D2WTFW.json","graph_json":"https://pith.science/api/pith-number/ZVR7TX7DXIPBUPDYNAF7D2WTFW/graph.json","events_json":"https://pith.science/api/pith-number/ZVR7TX7DXIPBUPDYNAF7D2WTFW/events.json","paper":"https://pith.science/paper/ZVR7TX7D"},"agent_actions":{"view_html":"https://pith.science/pith/ZVR7TX7DXIPBUPDYNAF7D2WTFW","download_json":"https://pith.science/pith/ZVR7TX7DXIPBUPDYNAF7D2WTFW.json","view_paper":"https://pith.science/paper/ZVR7TX7D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2105.11020&json=true","fetch_graph":"https://pith.science/api/pith-number/ZVR7TX7DXIPBUPDYNAF7D2WTFW/graph.json","fetch_events":"https://pith.science/api/pith-number/ZVR7TX7DXIPBUPDYNAF7D2WTFW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZVR7TX7DXIPBUPDYNAF7D2WTFW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZVR7TX7DXIPBUPDYNAF7D2WTFW/action/storage_attestation","attest_author":"https://pith.science/pith/ZVR7TX7DXIPBUPDYNAF7D2WTFW/action/author_attestation","sign_citation":"https://pith.science/pith/ZVR7TX7DXIPBUPDYNAF7D2WTFW/action/citation_signature","submit_replication":"https://pith.science/pith/ZVR7TX7DXIPBUPDYNAF7D2WTFW/action/replication_record"}},"created_at":"2026-05-22T01:03:38.414405+00:00","updated_at":"2026-05-22T01:03:38.414405+00:00"}