{
  "abstract": "We prove that, for either a Steinhaus random multiplicative function or a Rademacher random multiplicative function $f$, and every $\\varepsilon>0$, almost surely $|\\sum_{n\\le x}f(n)|\\ll_{\\varepsilon,f}\\sqrt{x}(\\log\\log x)^{1/4+\\varepsilon}$. Together with Harper's almost sure lower bound, this determines the sharp logarithmic exponent in both models. This settles Harper's conjecture on the large fluctuations of random multiplicative functions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.29429",
  "author_total": 2,
  "authors": [
    {
      "name": "Benjamin Durkan"
    },
    {
      "name": "Andrew Pearce-Crump"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/41f93yw598eg9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.29429",
  "fulltext_url": null,
  "id": "41f93yw598eg9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.29429",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7172274197",
  "pdf_url": "https://arxiv.org/pdf/2607.29429",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/41f93yw598eg9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2607.29429",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.29429"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A sharp almost sure upper bound for partial sums of random multiplicative functions",
  "title_text": "A sharp almost sure upper bound for partial sums of random multiplicative functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/41f93yw598eg9",
  "venue": "arXiv",
  "versions": []
}