{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:WAYRGLIC3BPEEW7TRZP7H2SYDS","short_pith_number":"pith:WAYRGLIC","schema_version":"1.0","canonical_sha256":"b031132d02d85e425bf38e5ff3ea581c9ca1fec0b188f24cbebd9338ab356837","source":{"kind":"arxiv","id":"1610.00511","version":1},"attestation_state":"computed","paper":{"title":"Ergodic averages with prime divisor weights in $L^{1}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.NT"],"primary_cat":"math.DS","authors_text":"Zoltan Buczolich","submitted_at":"2016-10-03T12:19:36Z","abstract_excerpt":"We show that $ { \\omega }(n)$ and $ { \\Omega }(n)$, the number of distinct prime factors of $n$ and the number of distinct prime factors of $n$ counted according to multiplicity are good weighting functions for the pointwise ergodic theorem in $L^{1}$. That is, if $g$ denotes one of these functions and $S_{g,K}=\\sum_{n\\leq K}g(n)$ then for every ergodic dynamical system $(X, { { \\cal A } },\\mu, { \\tau })$ and every $f\\in L^{1}(X)$ $$\\lim_{K\\to { \\infty }} \\frac{1}{S_{g,K}}\\sum_{n=1}^{K} g(n)f( { \\tau }^{n}x)=\\int_{X}fd\\mu \\text{ for $\\mu$ a.e. }x\\in X. $$\n  This answers a question raised by C."},"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":"1610.00511","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-10-03T12:19:36Z","cross_cats_sorted":["math.CA","math.NT"],"title_canon_sha256":"185a71cebdf2c43385a1562b0571e4ac9249df371fbaadc5001bd98df0c26f08","abstract_canon_sha256":"827892151287026a191320149cbc1d1ebe53051070bdfc692b26e1d458c9ecc3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:25.376647Z","signature_b64":"mLx+n8K0RONb76T5k3lsk7OjS33RrnVSI+6vjkuHDInuHaLQFPVtpIGY2GGZuwE7r5kS/FvSPQIq1yq+blAAAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b031132d02d85e425bf38e5ff3ea581c9ca1fec0b188f24cbebd9338ab356837","last_reissued_at":"2026-05-18T01:03:25.376194Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:25.376194Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ergodic averages with prime divisor weights in $L^{1}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.NT"],"primary_cat":"math.DS","authors_text":"Zoltan Buczolich","submitted_at":"2016-10-03T12:19:36Z","abstract_excerpt":"We show that $ { \\omega }(n)$ and $ { \\Omega }(n)$, the number of distinct prime factors of $n$ and the number of distinct prime factors of $n$ counted according to multiplicity are good weighting functions for the pointwise ergodic theorem in $L^{1}$. That is, if $g$ denotes one of these functions and $S_{g,K}=\\sum_{n\\leq K}g(n)$ then for every ergodic dynamical system $(X, { { \\cal A } },\\mu, { \\tau })$ and every $f\\in L^{1}(X)$ $$\\lim_{K\\to { \\infty }} \\frac{1}{S_{g,K}}\\sum_{n=1}^{K} g(n)f( { \\tau }^{n}x)=\\int_{X}fd\\mu \\text{ for $\\mu$ a.e. }x\\in X. $$\n  This answers a question raised by C."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.00511","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":"1610.00511","created_at":"2026-05-18T01:03:25.376260+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.00511v1","created_at":"2026-05-18T01:03:25.376260+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.00511","created_at":"2026-05-18T01:03:25.376260+00:00"},{"alias_kind":"pith_short_12","alias_value":"WAYRGLIC3BPE","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_16","alias_value":"WAYRGLIC3BPEEW7T","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_8","alias_value":"WAYRGLIC","created_at":"2026-05-18T12:30:48.956258+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/WAYRGLIC3BPEEW7TRZP7H2SYDS","json":"https://pith.science/pith/WAYRGLIC3BPEEW7TRZP7H2SYDS.json","graph_json":"https://pith.science/api/pith-number/WAYRGLIC3BPEEW7TRZP7H2SYDS/graph.json","events_json":"https://pith.science/api/pith-number/WAYRGLIC3BPEEW7TRZP7H2SYDS/events.json","paper":"https://pith.science/paper/WAYRGLIC"},"agent_actions":{"view_html":"https://pith.science/pith/WAYRGLIC3BPEEW7TRZP7H2SYDS","download_json":"https://pith.science/pith/WAYRGLIC3BPEEW7TRZP7H2SYDS.json","view_paper":"https://pith.science/paper/WAYRGLIC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.00511&json=true","fetch_graph":"https://pith.science/api/pith-number/WAYRGLIC3BPEEW7TRZP7H2SYDS/graph.json","fetch_events":"https://pith.science/api/pith-number/WAYRGLIC3BPEEW7TRZP7H2SYDS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WAYRGLIC3BPEEW7TRZP7H2SYDS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WAYRGLIC3BPEEW7TRZP7H2SYDS/action/storage_attestation","attest_author":"https://pith.science/pith/WAYRGLIC3BPEEW7TRZP7H2SYDS/action/author_attestation","sign_citation":"https://pith.science/pith/WAYRGLIC3BPEEW7TRZP7H2SYDS/action/citation_signature","submit_replication":"https://pith.science/pith/WAYRGLIC3BPEEW7TRZP7H2SYDS/action/replication_record"}},"created_at":"2026-05-18T01:03:25.376260+00:00","updated_at":"2026-05-18T01:03:25.376260+00:00"}