{"paper":{"title":"Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kaisa Matom\\\"aki, Maksym Radziwi{\\l}{\\l}, Terence Tao","submitted_at":"2017-07-05T10:55:31Z","abstract_excerpt":"We show that the expected asymptotic for the sums $\\sum_{X < n \\leq 2X} \\Lambda(n) \\Lambda(n+h)$, $\\sum_{X < n \\leq 2X} d_k(n) d_l(n+h)$, and $\\sum_{X < n \\leq 2X} \\Lambda(n) d_k(n+h)$ hold for almost all $h \\in [-H,H]$, provided that $X^{8/33+\\varepsilon} \\leq H \\leq X^{1-\\varepsilon}$, with an error term saving on average an arbitrary power of the logarithm over the trivial bound. Previous work of Mikawa, Perelli-Pintz and Baier-Browning-Marasingha-Zhao covered the range $H \\geq X^{1/3+\\varepsilon}$. We also obtain an analogous result for $\\sum_n \\Lambda(n) \\Lambda(N-n)$.\n  Our proof uses th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.01315","kind":"arxiv","version":3},"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"}