{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:6PYAAOG4M2KKWUBVI46E4BGOBN","short_pith_number":"pith:6PYAAOG4","schema_version":"1.0","canonical_sha256":"f3f00038dc6694ab5035473c4e04ce0b7097006ce2bd3cc97b2d91ac97da8a87","source":{"kind":"arxiv","id":"1907.04753","version":1},"attestation_state":"computed","paper":{"title":"Endpoint estimates for the maximal function over prime numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.NT"],"primary_cat":"math.DS","authors_text":"Bartosz Trojan","submitted_at":"2019-07-10T14:31:20Z","abstract_excerpt":"Given an ergodic dynamical system $(X, \\mathcal{B}, \\mu, T)$, we prove that for each function $f$ belonging to the Orlicz space $L(\\log L)^2(\\log \\log L)(X, \\mu)$, the ergodic averages \\[ \\frac{1}{\\pi(N)} \\sum_{p \\in \\mathbb{P}_N} f\\big(T^p x\\big), \\] converge for $\\mu$-almost all $x \\in X$, where $\\mathbb{P}_N$ is the set of prime numbers not larger that $N$ and $\\pi(N) = \\# \\mathbb{P}_N$."},"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":"1907.04753","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-07-10T14:31:20Z","cross_cats_sorted":["math.CA","math.NT"],"title_canon_sha256":"eb69c054768258cb8ed91934f85c98e9e1916f7a61769744f7fcbc5c2ef1315e","abstract_canon_sha256":"867b9671d0f6a257eb56e629a422a791db9599d0d35e56c563191ff1304dff35"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:56.858672Z","signature_b64":"lbjZ7X82GaNra6In3A9rbk6tBHpJTAdsSUeAo+VY4RQg7uXOP6H8nYzHqHqhLwUomaMdUnM82XSQrVVA13Q3CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f3f00038dc6694ab5035473c4e04ce0b7097006ce2bd3cc97b2d91ac97da8a87","last_reissued_at":"2026-05-17T23:40:56.857988Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:56.857988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Endpoint estimates for the maximal function over prime numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.NT"],"primary_cat":"math.DS","authors_text":"Bartosz Trojan","submitted_at":"2019-07-10T14:31:20Z","abstract_excerpt":"Given an ergodic dynamical system $(X, \\mathcal{B}, \\mu, T)$, we prove that for each function $f$ belonging to the Orlicz space $L(\\log L)^2(\\log \\log L)(X, \\mu)$, the ergodic averages \\[ \\frac{1}{\\pi(N)} \\sum_{p \\in \\mathbb{P}_N} f\\big(T^p x\\big), \\] converge for $\\mu$-almost all $x \\in X$, where $\\mathbb{P}_N$ is the set of prime numbers not larger that $N$ and $\\pi(N) = \\# \\mathbb{P}_N$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.04753","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":""},"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":"1907.04753","created_at":"2026-05-17T23:40:56.858112+00:00"},{"alias_kind":"arxiv_version","alias_value":"1907.04753v1","created_at":"2026-05-17T23:40:56.858112+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.04753","created_at":"2026-05-17T23:40:56.858112+00:00"},{"alias_kind":"pith_short_12","alias_value":"6PYAAOG4M2KK","created_at":"2026-05-18T12:33:10.108867+00:00"},{"alias_kind":"pith_short_16","alias_value":"6PYAAOG4M2KKWUBV","created_at":"2026-05-18T12:33:10.108867+00:00"},{"alias_kind":"pith_short_8","alias_value":"6PYAAOG4","created_at":"2026-05-18T12:33:10.108867+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/6PYAAOG4M2KKWUBVI46E4BGOBN","json":"https://pith.science/pith/6PYAAOG4M2KKWUBVI46E4BGOBN.json","graph_json":"https://pith.science/api/pith-number/6PYAAOG4M2KKWUBVI46E4BGOBN/graph.json","events_json":"https://pith.science/api/pith-number/6PYAAOG4M2KKWUBVI46E4BGOBN/events.json","paper":"https://pith.science/paper/6PYAAOG4"},"agent_actions":{"view_html":"https://pith.science/pith/6PYAAOG4M2KKWUBVI46E4BGOBN","download_json":"https://pith.science/pith/6PYAAOG4M2KKWUBVI46E4BGOBN.json","view_paper":"https://pith.science/paper/6PYAAOG4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1907.04753&json=true","fetch_graph":"https://pith.science/api/pith-number/6PYAAOG4M2KKWUBVI46E4BGOBN/graph.json","fetch_events":"https://pith.science/api/pith-number/6PYAAOG4M2KKWUBVI46E4BGOBN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6PYAAOG4M2KKWUBVI46E4BGOBN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6PYAAOG4M2KKWUBVI46E4BGOBN/action/storage_attestation","attest_author":"https://pith.science/pith/6PYAAOG4M2KKWUBVI46E4BGOBN/action/author_attestation","sign_citation":"https://pith.science/pith/6PYAAOG4M2KKWUBVI46E4BGOBN/action/citation_signature","submit_replication":"https://pith.science/pith/6PYAAOG4M2KKWUBVI46E4BGOBN/action/replication_record"}},"created_at":"2026-05-17T23:40:56.858112+00:00","updated_at":"2026-05-17T23:40:56.858112+00:00"}