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In this paper we establish the logarithmically averaged version $$ \\sum_{x/\\omega(x) < n \\leq x} \\frac{\\lambda(a_1 n + b_1) \\lambda(a_2 n+b_2)}{n} = o(\\log \\omega(x)) $$ of the Chowla conjecture as $x \\to \\infty$, where $1 \\leq \\omega(x) \\leq x$ is an arbitrary function of $x$ that goes to infinity as $x \\to \\infty$, "},"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":"1509.05422","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-17T20:17:58Z","cross_cats_sorted":[],"title_canon_sha256":"f801cd3002bf0df2d6a957033897abf33f4d7d31cb75eb637b0694586f14d618","abstract_canon_sha256":"37ba85f8765d4330b80ef90e2123aab5de14d27e60b133fe1b1bc6477818b853"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:18.318224Z","signature_b64":"NKeO9nJi8TtITZ/nIlH9LalswYVSy6UotZ02t2w6sB2mU73U6kEJrgiQnVJJpWHHgb3wpJ8YWIimvK5Im32BDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41e3412f23ba9abdd422e4e09a41fd8aac4926781443ac500d1f017a844e355d","last_reissued_at":"2026-05-18T01:10:18.317483Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:18.317483Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The logarithmically averaged Chowla and Elliott conjectures for two-point correlations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Terence Tao","submitted_at":"2015-09-17T20:17:58Z","abstract_excerpt":"Let $\\lambda$ denote the Liouville function. The Chowla conjecture, in the two-point correlation case, asserts that $$ \\sum_{n \\leq x} \\lambda(a_1 n + b_1) \\lambda(a_2 n+b_2) = o(x) $$ as $x \\to \\infty$, for any fixed natural numbers $a_1,a_2,b_1,b_2$ with $a_1b_2-a_2b_1 \\neq 0$. In this paper we establish the logarithmically averaged version $$ \\sum_{x/\\omega(x) < n \\leq x} \\frac{\\lambda(a_1 n + b_1) \\lambda(a_2 n+b_2)}{n} = o(\\log \\omega(x)) $$ of the Chowla conjecture as $x \\to \\infty$, where $1 \\leq \\omega(x) \\leq x$ is an arbitrary function of $x$ that goes to infinity as $x \\to \\infty$, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05422","kind":"arxiv","version":4},"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":"1509.05422","created_at":"2026-05-18T01:10:18.317593+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.05422v4","created_at":"2026-05-18T01:10:18.317593+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05422","created_at":"2026-05-18T01:10:18.317593+00:00"},{"alias_kind":"pith_short_12","alias_value":"IHRUCLZDXKNL","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_16","alias_value":"IHRUCLZDXKNL3VBC","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_8","alias_value":"IHRUCLZD","created_at":"2026-05-18T12:29:25.134429+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/IHRUCLZDXKNL3VBC4TQJUQP5RK","json":"https://pith.science/pith/IHRUCLZDXKNL3VBC4TQJUQP5RK.json","graph_json":"https://pith.science/api/pith-number/IHRUCLZDXKNL3VBC4TQJUQP5RK/graph.json","events_json":"https://pith.science/api/pith-number/IHRUCLZDXKNL3VBC4TQJUQP5RK/events.json","paper":"https://pith.science/paper/IHRUCLZD"},"agent_actions":{"view_html":"https://pith.science/pith/IHRUCLZDXKNL3VBC4TQJUQP5RK","download_json":"https://pith.science/pith/IHRUCLZDXKNL3VBC4TQJUQP5RK.json","view_paper":"https://pith.science/paper/IHRUCLZD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.05422&json=true","fetch_graph":"https://pith.science/api/pith-number/IHRUCLZDXKNL3VBC4TQJUQP5RK/graph.json","fetch_events":"https://pith.science/api/pith-number/IHRUCLZDXKNL3VBC4TQJUQP5RK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IHRUCLZDXKNL3VBC4TQJUQP5RK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IHRUCLZDXKNL3VBC4TQJUQP5RK/action/storage_attestation","attest_author":"https://pith.science/pith/IHRUCLZDXKNL3VBC4TQJUQP5RK/action/author_attestation","sign_citation":"https://pith.science/pith/IHRUCLZDXKNL3VBC4TQJUQP5RK/action/citation_signature","submit_replication":"https://pith.science/pith/IHRUCLZDXKNL3VBC4TQJUQP5RK/action/replication_record"}},"created_at":"2026-05-18T01:10:18.317593+00:00","updated_at":"2026-05-18T01:10:18.317593+00:00"}