{"paper":{"title":"Local Moments of M\\\"obius Fourier Polynomials and the Riemann Hypothesis","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.PR","authors_text":"Alberto Verjovsky","submitted_at":"2026-07-27T18:58:35Z","abstract_excerpt":"Let\n  $$ P_N(t)=\\frac1{\\sqrt N}\\sum_{n\\leq N}\\mu(n)\\e^{2\\pi i nt}, \\qquad t\\in\\T=\\mathbb R/\\mathbb Z. $$ We give a local probabilistic reformulation of the Riemann hypothesis by evaluating the normalized M\\\"obius Fourier polynomial \\(P_N\\) at a uniform random point in an arc of radius \\(c/N\\). We prove that RH is equivalent to subpolynomial growth of arbitrarily high finite local moments of these random variables. The principal quantitative tool is a local moment-to-point-value inequality which recovers the value \\(P_N(0)=M(N)/\\sqrt N\\) from local \\(L^q\\)-data (where $M$ denotes the Mertens fu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.25002","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/2607.25002/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"}