{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:6PYAAOG4M2KKWUBVI46E4BGOBN","short_pith_number":"pith:6PYAAOG4","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"},"canonical_sha256":"f3f00038dc6694ab5035473c4e04ce0b7097006ce2bd3cc97b2d91ac97da8a87","source":{"kind":"arxiv","id":"1907.04753","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.04753","created_at":"2026-05-17T23:40:56Z"},{"alias_kind":"arxiv_version","alias_value":"1907.04753v1","created_at":"2026-05-17T23:40:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.04753","created_at":"2026-05-17T23:40:56Z"},{"alias_kind":"pith_short_12","alias_value":"6PYAAOG4M2KK","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_16","alias_value":"6PYAAOG4M2KKWUBV","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_8","alias_value":"6PYAAOG4","created_at":"2026-05-18T12:33:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:6PYAAOG4M2KKWUBVI46E4BGOBN","target":"record","payload":{"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"},"canonical_sha256":"f3f00038dc6694ab5035473c4e04ce0b7097006ce2bd3cc97b2d91ac97da8a87","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"},"source_kind":"arxiv","source_id":"1907.04753","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:40:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ooyB6Kr9po5u/IGP3a0zxrVcpnK87p/ONO6oxpNWZmBAaXvoftplLG+riu1ilQX2K5JEp6n04dlnIb0fgrWdAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T10:39:37.129200Z"},"content_sha256":"41423dfc3016b99760167da46027d5573062c4ea5a06f0c21e0d577a081bca48","schema_version":"1.0","event_id":"sha256:41423dfc3016b99760167da46027d5573062c4ea5a06f0c21e0d577a081bca48"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:6PYAAOG4M2KKWUBVI46E4BGOBN","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:40:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NUBizn4aQqbmABOBxb/nBPf5Qdtf7OitELzUQ8MWwK5SPs1pwO/6+qTrni+uFyCEPh3LYEsJEzSifh5bNA51Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T10:39:37.136981Z"},"content_sha256":"5ab68fb56e3e4f341659a19974942fd95488eec5cca36050f4f91ba9e97a1b16","schema_version":"1.0","event_id":"sha256:5ab68fb56e3e4f341659a19974942fd95488eec5cca36050f4f91ba9e97a1b16"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6PYAAOG4M2KKWUBVI46E4BGOBN/bundle.json","state_url":"https://pith.science/pith/6PYAAOG4M2KKWUBVI46E4BGOBN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6PYAAOG4M2KKWUBVI46E4BGOBN/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-26T10:39:37Z","links":{"resolver":"https://pith.science/pith/6PYAAOG4M2KKWUBVI46E4BGOBN","bundle":"https://pith.science/pith/6PYAAOG4M2KKWUBVI46E4BGOBN/bundle.json","state":"https://pith.science/pith/6PYAAOG4M2KKWUBVI46E4BGOBN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6PYAAOG4M2KKWUBVI46E4BGOBN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:6PYAAOG4M2KKWUBVI46E4BGOBN","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"867b9671d0f6a257eb56e629a422a791db9599d0d35e56c563191ff1304dff35","cross_cats_sorted":["math.CA","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-07-10T14:31:20Z","title_canon_sha256":"eb69c054768258cb8ed91934f85c98e9e1916f7a61769744f7fcbc5c2ef1315e"},"schema_version":"1.0","source":{"id":"1907.04753","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.04753","created_at":"2026-05-17T23:40:56Z"},{"alias_kind":"arxiv_version","alias_value":"1907.04753v1","created_at":"2026-05-17T23:40:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.04753","created_at":"2026-05-17T23:40:56Z"},{"alias_kind":"pith_short_12","alias_value":"6PYAAOG4M2KK","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_16","alias_value":"6PYAAOG4M2KKWUBV","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_8","alias_value":"6PYAAOG4","created_at":"2026-05-18T12:33:10Z"}],"graph_snapshots":[{"event_id":"sha256:5ab68fb56e3e4f341659a19974942fd95488eec5cca36050f4f91ba9e97a1b16","target":"graph","created_at":"2026-05-17T23:40:56Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 $\\pi(N) = \\# \\mathbb{P}_N$.","authors_text":"Bartosz Trojan","cross_cats":["math.CA","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-07-10T14:31:20Z","title":"Endpoint estimates for the maximal function over prime numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.04753","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:41423dfc3016b99760167da46027d5573062c4ea5a06f0c21e0d577a081bca48","target":"record","created_at":"2026-05-17T23:40:56Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"867b9671d0f6a257eb56e629a422a791db9599d0d35e56c563191ff1304dff35","cross_cats_sorted":["math.CA","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-07-10T14:31:20Z","title_canon_sha256":"eb69c054768258cb8ed91934f85c98e9e1916f7a61769744f7fcbc5c2ef1315e"},"schema_version":"1.0","source":{"id":"1907.04753","kind":"arxiv","version":1}},"canonical_sha256":"f3f00038dc6694ab5035473c4e04ce0b7097006ce2bd3cc97b2d91ac97da8a87","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f3f00038dc6694ab5035473c4e04ce0b7097006ce2bd3cc97b2d91ac97da8a87","first_computed_at":"2026-05-17T23:40:56.857988Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:40:56.857988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lbjZ7X82GaNra6In3A9rbk6tBHpJTAdsSUeAo+VY4RQg7uXOP6H8nYzHqHqhLwUomaMdUnM82XSQrVVA13Q3CQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:40:56.858672Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.04753","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:41423dfc3016b99760167da46027d5573062c4ea5a06f0c21e0d577a081bca48","sha256:5ab68fb56e3e4f341659a19974942fd95488eec5cca36050f4f91ba9e97a1b16"],"state_sha256":"29c9f85ea315ad85cc990bfd1ce5d8542373ca816d69d49b479c8d9c2e08826e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qkR0QIHK9qx0IdIK5rtUEs8mbmuAuF1zwo08KVcxLU+gNrkzERh08GmpXMc2yf88+9hEZ1MncRHl2kx3iY/OAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T10:39:37.154581Z","bundle_sha256":"bf4b22b7795cca53cb8dab679c0c67afccd9c1d1e080a18db3d63a8cf0b7f3bb"}}