{"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"}