{"paper":{"title":"Fake Mu's","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Greg Martin, Michael J. Mossinghoff, Timothy S. Trudgian","submitted_at":"2021-12-09T21:54:34Z","abstract_excerpt":"Let $f(n)$ denote a multiplicative function with range $\\{-1,0,1\\}$, and let $F(x) = \\sum_{n\\leq x} f(n)$. Then $F(x)/\\sqrt{x} = a\\sqrt{x} + b + E(x)$, where $a$ and $b$ are constants and $E(x)$ is an error term that either tends to $0$ in the limit, or is expected to oscillate about $0$ in a roughly balanced manner. We say $F(x)$ has persistent bias $b$ (at the scale of $\\sqrt{x}$) in the first case, and apparent bias $b$ in the latter. For example, if $f(n)=\\mu(n)$, the M\\\"{o}bius function, then $F(x) = \\sum_{n\\leq x} \\mu(n)$ has $b=0$ so exhibits no persistent or apparent bias, while if $f("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.05227","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2112.05227/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}