{"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"}