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(\\ln x)^{b+1} \\; \\exp\\left(-c\\; \\sqrt{\\ln x}\\right)} < \\pi(x) < {\\mathrm{Li} \\over 1-a \\;(\\ln x)^{b+1} \\; \\exp\\left(-c\\; \\sqrt{\\ln x}\\right)}. \\] Furthermore, it is possible to establish exponentially tight effective upp"},"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":"2504.14458","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-04-20T02:35:53Z","cross_cats_sorted":[],"title_canon_sha256":"6d548dd8bc54afcc220aeccbce1770fe1d2ee9c93b2af93bd0f99e7b4c224246","abstract_canon_sha256":"44e0665b9a506ab71e64b2ac819ce76102ed3711e7ac9e98bc7005c7d26637fd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:21:45.747763Z","signature_b64":"BPiW98Y0qZeFsaVcPBBgzCQNA3mei95dt2Zj/T8W1pGn/dBsvQ+vPxynYdeZ9cnI0rMNPXfSfelcpbxG4hYADA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"88f40b49ca18fc536c21b4478329fb32c1b11fce339813c3a4204c7ec9b0aa3f","last_reissued_at":"2026-07-05T11:21:45.747206Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:21:45.747206Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The n-th prime exponentially","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Matt Visser (Victoria University of Wellington)","submitted_at":"2025-04-20T02:35:53Z","abstract_excerpt":"From known effective bounds on the prime counting function of the form \\[ |\\pi(x)-\\mathrm{Li}(x)| < a \\;x \\;(\\ln x)^{b} \\; \\exp\\left(-{c}\\; \\sqrt{\\ln x}\\right); \\qquad (x \\geq x_0); \\] it is possible to establish exponentially tight effective upper and lower bounds on the prime number theorem: For $x \\geq x_*$ where $x_*\\leq \\max\\{x_0,17\\}$ we have: \\[ {\\mathrm{Li} \\over 1+a\\; (\\ln x)^{b+1} \\; \\exp\\left(-c\\; \\sqrt{\\ln x}\\right)} < \\pi(x) < {\\mathrm{Li} \\over 1-a \\;(\\ln x)^{b+1} \\; \\exp\\left(-c\\; \\sqrt{\\ln x}\\right)}. \\] Furthermore, it is possible to establish exponentially tight effective upp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.14458","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/2504.14458/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":"2504.14458","created_at":"2026-07-05T11:21:45.747269+00:00"},{"alias_kind":"arxiv_version","alias_value":"2504.14458v1","created_at":"2026-07-05T11:21:45.747269+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.14458","created_at":"2026-07-05T11:21:45.747269+00:00"},{"alias_kind":"pith_short_12","alias_value":"RD2AWSOKDD6F","created_at":"2026-07-05T11:21:45.747269+00:00"},{"alias_kind":"pith_short_16","alias_value":"RD2AWSOKDD6FG3BB","created_at":"2026-07-05T11:21:45.747269+00:00"},{"alias_kind":"pith_short_8","alias_value":"RD2AWSOK","created_at":"2026-07-05T11:21:45.747269+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2507.14410","citing_title":"Behaviour of the sequence $\\vartheta_n = \\vartheta(p_n)$","ref_index":13,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RD2AWSOKDD6FG3BBWRDYGKP3GL","json":"https://pith.science/pith/RD2AWSOKDD6FG3BBWRDYGKP3GL.json","graph_json":"https://pith.science/api/pith-number/RD2AWSOKDD6FG3BBWRDYGKP3GL/graph.json","events_json":"https://pith.science/api/pith-number/RD2AWSOKDD6FG3BBWRDYGKP3GL/events.json","paper":"https://pith.science/paper/RD2AWSOK"},"agent_actions":{"view_html":"https://pith.science/pith/RD2AWSOKDD6FG3BBWRDYGKP3GL","download_json":"https://pith.science/pith/RD2AWSOKDD6FG3BBWRDYGKP3GL.json","view_paper":"https://pith.science/paper/RD2AWSOK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2504.14458&json=true","fetch_graph":"https://pith.science/api/pith-number/RD2AWSOKDD6FG3BBWRDYGKP3GL/graph.json","fetch_events":"https://pith.science/api/pith-number/RD2AWSOKDD6FG3BBWRDYGKP3GL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RD2AWSOKDD6FG3BBWRDYGKP3GL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RD2AWSOKDD6FG3BBWRDYGKP3GL/action/storage_attestation","attest_author":"https://pith.science/pith/RD2AWSOKDD6FG3BBWRDYGKP3GL/action/author_attestation","sign_citation":"https://pith.science/pith/RD2AWSOKDD6FG3BBWRDYGKP3GL/action/citation_signature","submit_replication":"https://pith.science/pith/RD2AWSOKDD6FG3BBWRDYGKP3GL/action/replication_record"}},"created_at":"2026-07-05T11:21:45.747269+00:00","updated_at":"2026-07-05T11:21:45.747269+00:00"}