{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:5EP2JFSZJG5MSMJMMUPQX5MJLI","short_pith_number":"pith:5EP2JFSZ","schema_version":"1.0","canonical_sha256":"e91fa4965949bac9312c651f0bf5895a18d7b34dafba99c9815852e6526f6bcd","source":{"kind":"arxiv","id":"2505.07352","version":2},"attestation_state":"computed","paper":{"title":"Brownian behaviour of the Riemann zeta function around the critical line","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Louis Vassaux","submitted_at":"2025-05-12T08:44:45Z","abstract_excerpt":"We establish a Brownian extension to Selberg's central limit theorem for the Riemann zeta function. This implies various limiting distributions for $\\zeta$, including an analogue of the reflection principle for the maximum of the Brownian motion: as $T$ diverges, for any $u>0$ we have \\[ \\frac{1}{T}\\cdot {\\rm meas}\\Big\\{0\\leq t\\leq T:\\max_{\\sigma\\geq \\tfrac{1}{2}}\\log|\\zeta(\\sigma+i t)|\\geq u \\sqrt{\\tfrac{1}{2}\\log \\log T} \\Big\\}\\to 2 \\displaystyle\\int_u^{\\infty} \\frac{e^{-\\frac{x^2}{2}}}{\\sqrt{2\\pi}}\\mathrm{d} x. \\]"},"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":"2505.07352","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-05-12T08:44:45Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"1bf38221fe9b80f67a432f6c7d3da6f86a6be35d7b362534196f2ade63d7f4a6","abstract_canon_sha256":"47e5ccf3d5b39a4cf0e97a6b63aafd5a3732ab49bce0246274bbc01fc73179a2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T03:04:32.423434Z","signature_b64":"Q004KDmkAG7C3LL+HUCKe7TTPSKIPdG3CUwXJtxwPilCxP6kvPvuSmWG3qv63eiECl6Qtkb95clYfW+PvxOoBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e91fa4965949bac9312c651f0bf5895a18d7b34dafba99c9815852e6526f6bcd","last_reissued_at":"2026-06-02T03:04:32.422991Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T03:04:32.422991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Brownian behaviour of the Riemann zeta function around the critical line","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Louis Vassaux","submitted_at":"2025-05-12T08:44:45Z","abstract_excerpt":"We establish a Brownian extension to Selberg's central limit theorem for the Riemann zeta function. This implies various limiting distributions for $\\zeta$, including an analogue of the reflection principle for the maximum of the Brownian motion: as $T$ diverges, for any $u>0$ we have \\[ \\frac{1}{T}\\cdot {\\rm meas}\\Big\\{0\\leq t\\leq T:\\max_{\\sigma\\geq \\tfrac{1}{2}}\\log|\\zeta(\\sigma+i t)|\\geq u \\sqrt{\\tfrac{1}{2}\\log \\log T} \\Big\\}\\to 2 \\displaystyle\\int_u^{\\infty} \\frac{e^{-\\frac{x^2}{2}}}{\\sqrt{2\\pi}}\\mathrm{d} x. \\]"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.07352","kind":"arxiv","version":2},"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/2505.07352/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":"2505.07352","created_at":"2026-06-02T03:04:32.423056+00:00"},{"alias_kind":"arxiv_version","alias_value":"2505.07352v2","created_at":"2026-06-02T03:04:32.423056+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.07352","created_at":"2026-06-02T03:04:32.423056+00:00"},{"alias_kind":"pith_short_12","alias_value":"5EP2JFSZJG5M","created_at":"2026-06-02T03:04:32.423056+00:00"},{"alias_kind":"pith_short_16","alias_value":"5EP2JFSZJG5MSMJM","created_at":"2026-06-02T03:04:32.423056+00:00"},{"alias_kind":"pith_short_8","alias_value":"5EP2JFSZ","created_at":"2026-06-02T03:04:32.423056+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/5EP2JFSZJG5MSMJMMUPQX5MJLI","json":"https://pith.science/pith/5EP2JFSZJG5MSMJMMUPQX5MJLI.json","graph_json":"https://pith.science/api/pith-number/5EP2JFSZJG5MSMJMMUPQX5MJLI/graph.json","events_json":"https://pith.science/api/pith-number/5EP2JFSZJG5MSMJMMUPQX5MJLI/events.json","paper":"https://pith.science/paper/5EP2JFSZ"},"agent_actions":{"view_html":"https://pith.science/pith/5EP2JFSZJG5MSMJMMUPQX5MJLI","download_json":"https://pith.science/pith/5EP2JFSZJG5MSMJMMUPQX5MJLI.json","view_paper":"https://pith.science/paper/5EP2JFSZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2505.07352&json=true","fetch_graph":"https://pith.science/api/pith-number/5EP2JFSZJG5MSMJMMUPQX5MJLI/graph.json","fetch_events":"https://pith.science/api/pith-number/5EP2JFSZJG5MSMJMMUPQX5MJLI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5EP2JFSZJG5MSMJMMUPQX5MJLI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5EP2JFSZJG5MSMJMMUPQX5MJLI/action/storage_attestation","attest_author":"https://pith.science/pith/5EP2JFSZJG5MSMJMMUPQX5MJLI/action/author_attestation","sign_citation":"https://pith.science/pith/5EP2JFSZJG5MSMJMMUPQX5MJLI/action/citation_signature","submit_replication":"https://pith.science/pith/5EP2JFSZJG5MSMJMMUPQX5MJLI/action/replication_record"}},"created_at":"2026-06-02T03:04:32.423056+00:00","updated_at":"2026-06-02T03:04:32.423056+00:00"}