{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:AJRARHW5IL3YM3DOVDV23NFHPU","short_pith_number":"pith:AJRARHW5","schema_version":"1.0","canonical_sha256":"0262089edd42f7866c6ea8ebadb4a77d29c0ab48db2cf944bad60661e3978df8","source":{"kind":"arxiv","id":"1605.04628","version":1},"attestation_state":"computed","paper":{"title":"Equivalence of the logarithmically averaged Chowla and Sarnak conjectures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Terence Tao","submitted_at":"2016-05-16T01:39:46Z","abstract_excerpt":"Let $\\lambda$ denote the Liouville function. The Chowla conjecture asserts that $$ \\sum_{n \\leq X} \\lambda(a_1 n + b_1) \\lambda(a_2 n+b_2) \\dots \\lambda(a_k n + b_k) = o_{X \\to \\infty}(X) $$ for any fixed natural numbers $a_1,a_2,\\dots,a_k$ and non-negative integer $b_1,b_2,\\dots,b_k$ with $a_ib_j-a_jb_i \\neq 0$ for all $1 \\leq i < j \\leq k$, and any $X \\geq 1$. This conjecture is open for $k \\geq 2$. As is well known, this conjecture implies the conjecture of Sarnak that $$ \\sum_{n \\leq X} \\lambda(n) f(n) = o_{X \\to \\infty}(X)$$ whenever $f : {\\bf N} \\to {\\bf C}$ is a fixed deterministic sequ"},"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":"1605.04628","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-05-16T01:39:46Z","cross_cats_sorted":[],"title_canon_sha256":"fb97ed8c7cfc4e5002b4b5bd2c436bd0aa78a2e85e7b2d5d7b206a3ea21ec75d","abstract_canon_sha256":"3191bad4087926d621130e153647d2832abe64aad7cd417d907450a58d458c79"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:14:46.686335Z","signature_b64":"aOmDFXboPI9pmwNTSsKm/6J4SiG9bOavTzGLauxmUK/xFjCTNIPar/H9M9LtNmH4Hc94xreeYcT30qSyWXhdBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0262089edd42f7866c6ea8ebadb4a77d29c0ab48db2cf944bad60661e3978df8","last_reissued_at":"2026-05-18T01:14:46.685653Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:14:46.685653Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Equivalence of the logarithmically averaged Chowla and Sarnak conjectures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Terence Tao","submitted_at":"2016-05-16T01:39:46Z","abstract_excerpt":"Let $\\lambda$ denote the Liouville function. The Chowla conjecture asserts that $$ \\sum_{n \\leq X} \\lambda(a_1 n + b_1) \\lambda(a_2 n+b_2) \\dots \\lambda(a_k n + b_k) = o_{X \\to \\infty}(X) $$ for any fixed natural numbers $a_1,a_2,\\dots,a_k$ and non-negative integer $b_1,b_2,\\dots,b_k$ with $a_ib_j-a_jb_i \\neq 0$ for all $1 \\leq i < j \\leq k$, and any $X \\geq 1$. This conjecture is open for $k \\geq 2$. As is well known, this conjecture implies the conjecture of Sarnak that $$ \\sum_{n \\leq X} \\lambda(n) f(n) = o_{X \\to \\infty}(X)$$ whenever $f : {\\bf N} \\to {\\bf C}$ is a fixed deterministic sequ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.04628","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":"1605.04628","created_at":"2026-05-18T01:14:46.685765+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.04628v1","created_at":"2026-05-18T01:14:46.685765+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.04628","created_at":"2026-05-18T01:14:46.685765+00:00"},{"alias_kind":"pith_short_12","alias_value":"AJRARHW5IL3Y","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_16","alias_value":"AJRARHW5IL3YM3DO","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_8","alias_value":"AJRARHW5","created_at":"2026-05-18T12:30:07.202191+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/AJRARHW5IL3YM3DOVDV23NFHPU","json":"https://pith.science/pith/AJRARHW5IL3YM3DOVDV23NFHPU.json","graph_json":"https://pith.science/api/pith-number/AJRARHW5IL3YM3DOVDV23NFHPU/graph.json","events_json":"https://pith.science/api/pith-number/AJRARHW5IL3YM3DOVDV23NFHPU/events.json","paper":"https://pith.science/paper/AJRARHW5"},"agent_actions":{"view_html":"https://pith.science/pith/AJRARHW5IL3YM3DOVDV23NFHPU","download_json":"https://pith.science/pith/AJRARHW5IL3YM3DOVDV23NFHPU.json","view_paper":"https://pith.science/paper/AJRARHW5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.04628&json=true","fetch_graph":"https://pith.science/api/pith-number/AJRARHW5IL3YM3DOVDV23NFHPU/graph.json","fetch_events":"https://pith.science/api/pith-number/AJRARHW5IL3YM3DOVDV23NFHPU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AJRARHW5IL3YM3DOVDV23NFHPU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AJRARHW5IL3YM3DOVDV23NFHPU/action/storage_attestation","attest_author":"https://pith.science/pith/AJRARHW5IL3YM3DOVDV23NFHPU/action/author_attestation","sign_citation":"https://pith.science/pith/AJRARHW5IL3YM3DOVDV23NFHPU/action/citation_signature","submit_replication":"https://pith.science/pith/AJRARHW5IL3YM3DOVDV23NFHPU/action/replication_record"}},"created_at":"2026-05-18T01:14:46.685765+00:00","updated_at":"2026-05-18T01:14:46.685765+00:00"}