{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:WAYRGLIC3BPEEW7TRZP7H2SYDS","short_pith_number":"pith:WAYRGLIC","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"},"canonical_sha256":"b031132d02d85e425bf38e5ff3ea581c9ca1fec0b188f24cbebd9338ab356837","source":{"kind":"arxiv","id":"1610.00511","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.00511","created_at":"2026-05-18T01:03:25Z"},{"alias_kind":"arxiv_version","alias_value":"1610.00511v1","created_at":"2026-05-18T01:03:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.00511","created_at":"2026-05-18T01:03:25Z"},{"alias_kind":"pith_short_12","alias_value":"WAYRGLIC3BPE","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"WAYRGLIC3BPEEW7T","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"WAYRGLIC","created_at":"2026-05-18T12:30:48Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:WAYRGLIC3BPEEW7TRZP7H2SYDS","target":"record","payload":{"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"},"canonical_sha256":"b031132d02d85e425bf38e5ff3ea581c9ca1fec0b188f24cbebd9338ab356837","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"},"source_kind":"arxiv","source_id":"1610.00511","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":"2026-05-18T01:03:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"67GOv9mBm1PKxmBAlk0i4w4gy/z0ETdpqrkJDfgcWhcnElxS73PuAg03NTNf48dHmiH8JgxZWkzGNjyZmFjBBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T08:41:37.248794Z"},"content_sha256":"c2adffe3f5c0318e5067c37548c123c3e632311c9b288fefa781e14a73a89319","schema_version":"1.0","event_id":"sha256:c2adffe3f5c0318e5067c37548c123c3e632311c9b288fefa781e14a73a89319"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:WAYRGLIC3BPEEW7TRZP7H2SYDS","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:03:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Sss9sWUCJjJxHOKMNKhb4fu0dM0xnD/isB3/6BVTdTcquqOPuJufKr1DQuzafWVcGXYBGjvtDEEEsQbLlR9oBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T08:41:37.249145Z"},"content_sha256":"311f3bdcee820074a482ad0379967afb3c60d343ec577fd3760bf6d8f0a2d40d","schema_version":"1.0","event_id":"sha256:311f3bdcee820074a482ad0379967afb3c60d343ec577fd3760bf6d8f0a2d40d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WAYRGLIC3BPEEW7TRZP7H2SYDS/bundle.json","state_url":"https://pith.science/pith/WAYRGLIC3BPEEW7TRZP7H2SYDS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WAYRGLIC3BPEEW7TRZP7H2SYDS/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":"2026-06-23T08:41:37Z","links":{"resolver":"https://pith.science/pith/WAYRGLIC3BPEEW7TRZP7H2SYDS","bundle":"https://pith.science/pith/WAYRGLIC3BPEEW7TRZP7H2SYDS/bundle.json","state":"https://pith.science/pith/WAYRGLIC3BPEEW7TRZP7H2SYDS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WAYRGLIC3BPEEW7TRZP7H2SYDS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:WAYRGLIC3BPEEW7TRZP7H2SYDS","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":"827892151287026a191320149cbc1d1ebe53051070bdfc692b26e1d458c9ecc3","cross_cats_sorted":["math.CA","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-10-03T12:19:36Z","title_canon_sha256":"185a71cebdf2c43385a1562b0571e4ac9249df371fbaadc5001bd98df0c26f08"},"schema_version":"1.0","source":{"id":"1610.00511","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.00511","created_at":"2026-05-18T01:03:25Z"},{"alias_kind":"arxiv_version","alias_value":"1610.00511v1","created_at":"2026-05-18T01:03:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.00511","created_at":"2026-05-18T01:03:25Z"},{"alias_kind":"pith_short_12","alias_value":"WAYRGLIC3BPE","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"WAYRGLIC3BPEEW7T","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"WAYRGLIC","created_at":"2026-05-18T12:30:48Z"}],"graph_snapshots":[{"event_id":"sha256:311f3bdcee820074a482ad0379967afb3c60d343ec577fd3760bf6d8f0a2d40d","target":"graph","created_at":"2026-05-18T01:03:25Z","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":"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.","authors_text":"Zoltan Buczolich","cross_cats":["math.CA","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-10-03T12:19:36Z","title":"Ergodic averages with prime divisor weights in $L^{1}$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.00511","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:c2adffe3f5c0318e5067c37548c123c3e632311c9b288fefa781e14a73a89319","target":"record","created_at":"2026-05-18T01:03:25Z","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":"827892151287026a191320149cbc1d1ebe53051070bdfc692b26e1d458c9ecc3","cross_cats_sorted":["math.CA","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-10-03T12:19:36Z","title_canon_sha256":"185a71cebdf2c43385a1562b0571e4ac9249df371fbaadc5001bd98df0c26f08"},"schema_version":"1.0","source":{"id":"1610.00511","kind":"arxiv","version":1}},"canonical_sha256":"b031132d02d85e425bf38e5ff3ea581c9ca1fec0b188f24cbebd9338ab356837","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b031132d02d85e425bf38e5ff3ea581c9ca1fec0b188f24cbebd9338ab356837","first_computed_at":"2026-05-18T01:03:25.376194Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:03:25.376194Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mLx+n8K0RONb76T5k3lsk7OjS33RrnVSI+6vjkuHDInuHaLQFPVtpIGY2GGZuwE7r5kS/FvSPQIq1yq+blAAAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:03:25.376647Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.00511","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c2adffe3f5c0318e5067c37548c123c3e632311c9b288fefa781e14a73a89319","sha256:311f3bdcee820074a482ad0379967afb3c60d343ec577fd3760bf6d8f0a2d40d"],"state_sha256":"971bb80a74d3217b2373a03e69c103bed4c979779caa3a2033932984072410ce"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pfhP7HuuNxYMj5Q1nOCibblEuxoLzjBuEyeu2tXgB2zs1/MzU/dTHKEC63HxHEp/Rx7BUc6Q/aLHjDnIBMKUCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T08:41:37.251195Z","bundle_sha256":"262f2ab3aaf7a8cc8274f9c06a3d62e4756af3184420112eb37d053fa97e5117"}}