{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:NPGMCVPWKNN2D3WRNS25FOEFQC","short_pith_number":"pith:NPGMCVPW","canonical_record":{"source":{"id":"0908.0680","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-08-05T15:11:46Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"279b9133a45b27ed24f2df6457fa7c2fab5c79189de9971d930f9d1d85744ae5","abstract_canon_sha256":"7b1ccf9c1bc98e79df48cbeaa7dac5f09a093a84862ae4ed3086133c1e0f46da"},"schema_version":"1.0"},"canonical_sha256":"6bccc155f6535ba1eed16cb5d2b885809170672c95ea8ad07ae71b2f56664a2e","source":{"kind":"arxiv","id":"0908.0680","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.0680","created_at":"2026-05-18T00:40:29Z"},{"alias_kind":"arxiv_version","alias_value":"0908.0680v1","created_at":"2026-05-18T00:40:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.0680","created_at":"2026-05-18T00:40:29Z"},{"alias_kind":"pith_short_12","alias_value":"NPGMCVPWKNN2","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"NPGMCVPWKNN2D3WR","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"NPGMCVPW","created_at":"2026-05-18T12:26:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:NPGMCVPWKNN2D3WRNS25FOEFQC","target":"record","payload":{"canonical_record":{"source":{"id":"0908.0680","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-08-05T15:11:46Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"279b9133a45b27ed24f2df6457fa7c2fab5c79189de9971d930f9d1d85744ae5","abstract_canon_sha256":"7b1ccf9c1bc98e79df48cbeaa7dac5f09a093a84862ae4ed3086133c1e0f46da"},"schema_version":"1.0"},"canonical_sha256":"6bccc155f6535ba1eed16cb5d2b885809170672c95ea8ad07ae71b2f56664a2e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:29.116646Z","signature_b64":"d2k+fQAMXPFxk3H9sT8bAnV6XieKyV9bagjAriBsmVnczc0LLVUpSoUbAbiBBwYJmnHg8rGbNF9bWxTz1pY0AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6bccc155f6535ba1eed16cb5d2b885809170672c95ea8ad07ae71b2f56664a2e","last_reissued_at":"2026-05-18T00:40:29.116149Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:29.116149Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0908.0680","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":"2026-05-18T00:40:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xGC7sIcIoKS/YfJ+4GE6SnF2/EsBM64ak04K+bpZph+2kupiJNZ2KD++wIvU8HlrdjS5ElSsBikcaPm3Q8wZCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T21:31:02.354360Z"},"content_sha256":"00240efc05e2db78383889b9a50dbf68e917574511047c474e46d0ba401aa561","schema_version":"1.0","event_id":"sha256:00240efc05e2db78383889b9a50dbf68e917574511047c474e46d0ba401aa561"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:NPGMCVPWKNN2D3WRNS25FOEFQC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A general strong law of large numbers for additive arithmetic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Istvan Berkes, Michel Weber","submitted_at":"2009-08-05T15:11:46Z","abstract_excerpt":"Let $f(n)$ be a strongly additive complex valued arithmetic function. Under mild conditions on $f$, we prove the following weighted strong law of large numbers: if $ X,X_1,X_2,... $ is any sequence of integrable i.i.d. random variables, then $$ \\lim_{N\\to \\infty} {\\sum_{n=1}^N f(n) X_n \\over\\sum_{n=1}^N f(n)} \\buildrel{a.s.}\\over{=} \\E X . $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.0680","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":"2026-05-18T00:40:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Gyh01b19/71rD+v9CBwwaVAODsAlWDlg3/5IAd22UbeaonDqfI8wwEWpEMVVfUGIWRRlnTQpTB7wWzqxfXu0Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T21:31:02.354707Z"},"content_sha256":"792269d54770a2de295a2ec7edc87c65c749ca937c0627c3a2c63fba4b61e3e1","schema_version":"1.0","event_id":"sha256:792269d54770a2de295a2ec7edc87c65c749ca937c0627c3a2c63fba4b61e3e1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NPGMCVPWKNN2D3WRNS25FOEFQC/bundle.json","state_url":"https://pith.science/pith/NPGMCVPWKNN2D3WRNS25FOEFQC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NPGMCVPWKNN2D3WRNS25FOEFQC/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":"2026-06-01T21:31:02Z","links":{"resolver":"https://pith.science/pith/NPGMCVPWKNN2D3WRNS25FOEFQC","bundle":"https://pith.science/pith/NPGMCVPWKNN2D3WRNS25FOEFQC/bundle.json","state":"https://pith.science/pith/NPGMCVPWKNN2D3WRNS25FOEFQC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NPGMCVPWKNN2D3WRNS25FOEFQC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:NPGMCVPWKNN2D3WRNS25FOEFQC","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":"7b1ccf9c1bc98e79df48cbeaa7dac5f09a093a84862ae4ed3086133c1e0f46da","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-08-05T15:11:46Z","title_canon_sha256":"279b9133a45b27ed24f2df6457fa7c2fab5c79189de9971d930f9d1d85744ae5"},"schema_version":"1.0","source":{"id":"0908.0680","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.0680","created_at":"2026-05-18T00:40:29Z"},{"alias_kind":"arxiv_version","alias_value":"0908.0680v1","created_at":"2026-05-18T00:40:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.0680","created_at":"2026-05-18T00:40:29Z"},{"alias_kind":"pith_short_12","alias_value":"NPGMCVPWKNN2","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"NPGMCVPWKNN2D3WR","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"NPGMCVPW","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:792269d54770a2de295a2ec7edc87c65c749ca937c0627c3a2c63fba4b61e3e1","target":"graph","created_at":"2026-05-18T00:40:29Z","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":"Let $f(n)$ be a strongly additive complex valued arithmetic function. Under mild conditions on $f$, we prove the following weighted strong law of large numbers: if $ X,X_1,X_2,... $ is any sequence of integrable i.i.d. random variables, then $$ \\lim_{N\\to \\infty} {\\sum_{n=1}^N f(n) X_n \\over\\sum_{n=1}^N f(n)} \\buildrel{a.s.}\\over{=} \\E X . $$","authors_text":"Istvan Berkes, Michel Weber","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-08-05T15:11:46Z","title":"A general strong law of large numbers for additive arithmetic functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.0680","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":"sha256:00240efc05e2db78383889b9a50dbf68e917574511047c474e46d0ba401aa561","target":"record","created_at":"2026-05-18T00:40:29Z","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":"7b1ccf9c1bc98e79df48cbeaa7dac5f09a093a84862ae4ed3086133c1e0f46da","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-08-05T15:11:46Z","title_canon_sha256":"279b9133a45b27ed24f2df6457fa7c2fab5c79189de9971d930f9d1d85744ae5"},"schema_version":"1.0","source":{"id":"0908.0680","kind":"arxiv","version":1}},"canonical_sha256":"6bccc155f6535ba1eed16cb5d2b885809170672c95ea8ad07ae71b2f56664a2e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6bccc155f6535ba1eed16cb5d2b885809170672c95ea8ad07ae71b2f56664a2e","first_computed_at":"2026-05-18T00:40:29.116149Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:40:29.116149Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"d2k+fQAMXPFxk3H9sT8bAnV6XieKyV9bagjAriBsmVnczc0LLVUpSoUbAbiBBwYJmnHg8rGbNF9bWxTz1pY0AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:40:29.116646Z","signed_message":"canonical_sha256_bytes"},"source_id":"0908.0680","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:00240efc05e2db78383889b9a50dbf68e917574511047c474e46d0ba401aa561","sha256:792269d54770a2de295a2ec7edc87c65c749ca937c0627c3a2c63fba4b61e3e1"],"state_sha256":"ed2f23f7646afccba797a8cf75d144ded908baed299eb4a607149d7e408bc2d7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2L4VHmOM6yID2m+mkzY7J/12n0Dw4ZQ5T0aq+h9ncAPUVYhRm7IC/3AZB+G8Zwwx5jdlOs85pFWfWgysL8oZBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T21:31:02.356628Z","bundle_sha256":"4f2f452d8090b835b81553a1a03e04a6e4482f0b005f801ad99ffd60ae87db2c"}}