{"paper":{"title":"Moments of random multiplicative functions, I: Low moments, better than squareroot cancellation, and critical multiplicative chaos","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.FA","math.PR"],"primary_cat":"math.NT","authors_text":"Adam J. Harper","submitted_at":"2017-03-20T10:20:02Z","abstract_excerpt":"We determine the order of magnitude of $\\mathbb{E}|\\sum_{n \\leq x} f(n)|^{2q}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, and $0 \\leq q \\leq 1$. In the Steinhaus case, this is equivalent to determining the order of $\\lim_{T \\rightarrow \\infty} \\frac{1}{T} \\int_{0}^{T} |\\sum_{n \\leq x} n^{-it}|^{2q} dt$.\n  In particular, we find that $\\mathbb{E}|\\sum_{n \\leq x} f(n)| \\asymp \\sqrt{x}/(\\log\\log x)^{1/4}$. This proves a conjecture of Helson that one should have better than squareroot cancellation in the first moment, and disproves counter-conjectures of various othe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.06654","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"}