{"paper":{"title":"A smooth shift approach for a Ramanujan expansion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Giovanni Coppola","submitted_at":"2019-01-06T18:10:06Z","abstract_excerpt":"All arithmetical functions $F$ satisfying Ramanujan Conjecture, i.e., $F(n)\\ll_{\\varepsilon}n^{\\varepsilon}$, and with $Q-$smooth divisors, i.e., with Eratosthenes transform $F':=F\\ast \\mu$ supported in $Q-$smooth numbers, have a kind of unique Ramanujan expansion; also, these Ramanujan coefficients decay very well to $0$ and have two explicit expressions (in the style of Carmichael and Wintner). This general result, then, is applied to the shift-Ramanujan expansions, i.e., the expansions for correlations with respect to the shift, whence the title."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.01584","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"}