{
  "abstract": "Let $f$ be an extended Rademacher random multiplicative function (RMF). We show that, for every fixed deterministic function $V(x)$ tending to infinity, almost surely there are arbitrarily large $x$ for which \\[ \\sum_{n\\leq x}f(n) \\geq \\frac{\\sqrt{x}(\\log\\log x)^{1/4}}{V(x)}. \\] The corresponding negative fluctuation holds as well. In particular, this gives an affirmative answer to Erdős Problem~\\#1144. As a byproduct, our result has a direct corollary giving new almost sure lower bounds $\\log\\log x$ on the number of sign changes of partial sums up to $x$ for all sufficiently large $x$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12940",
  "author_total": 2,
  "authors": [
    {
      "name": "Haozhe Gou"
    },
    {
      "name": "Max Wenqiang Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5a6z2ty9ma2hy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12940",
  "fulltext_url": null,
  "id": "5a6z2ty9ma2hy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12940",
  "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": "W7212540852",
  "pdf_url": "https://arxiv.org/pdf/2609.12940",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5a6z2ty9ma2hy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.12940",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12940"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Large fluctuations of extended Rademacher random multiplicative functions",
  "title_text": "Large fluctuations of extended Rademacher random multiplicative functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/5a6z2ty9ma2hy",
  "venue": "arXiv",
  "versions": []
}