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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 $x \\to \\infty$, ","authors_text":"Terence Tao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-17T20:17:58Z","title":"The logarithmically averaged Chowla and Elliott conjectures for two-point correlations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05422","kind":"arxiv","version":4},"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":"sha256:361121c78f7c89b70501a83e0e04570e9d7099c52545973ecc93a339c9448ab5","target":"record","created_at":"2026-05-18T01:10:18Z","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":"37ba85f8765d4330b80ef90e2123aab5de14d27e60b133fe1b1bc6477818b853","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-17T20:17:58Z","title_canon_sha256":"f801cd3002bf0df2d6a957033897abf33f4d7d31cb75eb637b0694586f14d618"},"schema_version":"1.0","source":{"id":"1509.05422","kind":"arxiv","version":4}},"canonical_sha256":"41e3412f23ba9abdd422e4e09a41fd8aac4926781443ac500d1f017a844e355d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"41e3412f23ba9abdd422e4e09a41fd8aac4926781443ac500d1f017a844e355d","first_computed_at":"2026-05-18T01:10:18.317483Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:10:18.317483Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NKeO9nJi8TtITZ/nIlH9LalswYVSy6UotZ02t2w6sB2mU73U6kEJrgiQnVJJpWHHgb3wpJ8YWIimvK5Im32BDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:10:18.318224Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.05422","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:361121c78f7c89b70501a83e0e04570e9d7099c52545973ecc93a339c9448ab5","sha256:45efa6aa9264d8af210b523c4e6a587d641ad950c586a0a24efcd74b4bbd7cc4"],"state_sha256":"259a3352386b5dab26577ef4746cc13291fda8fff4dbfbf76b9eb761af164498"}