{
  "abstract": "Let $f$ be a Rademacher random multiplicative function supported on the squarefree integers, and let $M(x)=\\sum_{n\\le x}f(n)$. We prove a two-sided law of the iterated logarithm normalized by the full predictable quadratic variation \\[ Q(x)=\\sum_{p\\le x}\\bigg(\\sum_{\\substack{m\\le x/p\\\\ P(m)<p}}f(m)\\bigg)^2 \\] of the martingale obtained by revealing the prime signs. Almost surely, \\[ \\limsup_{x\\to\\infty}\\frac{M(x)}{\\sqrt{2Q(x)\\log\\log x}}=1, \\qquad \\liminf_{x\\to\\infty}\\frac{M(x)}{\\sqrt{2Q(x)\\log\\log x}}=-1. \\] The same equalities hold with $\\log\\log Q(x)$ in the denominator. The almost sure cluster set of the normalized sum is $[-1,1]$. This gives a self-normalized version of the questions of Wintner and Erdős on almost sure fluctuations. We also prove that $Q(x)/q(x)$ cannot converge in probability to $1$ for any positive deterministic function $q$. Moreover, \\[ \\limsup_{x\\to\\infty} \\frac{|M(x)|}{\\sqrt{x}(\\log\\log x)^{1/4}} \\] is either an almost surely finite and positive random variable with nondegenerate distribution or infinite almost surely. We do not determine which alternative holds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40605",
  "author_total": 2,
  "authors": [
    {
      "name": "Dariusz Buraczewski"
    },
    {
      "name": "Alexander Iksanov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/595ngaq4cw3xt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40605",
  "fulltext_url": null,
  "id": "595ngaq4cw3xt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40605",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40605",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/595ngaq4cw3xt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:35Z",
      "native_id": "2610.40605",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40605"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A self-normalized law of the iterated logarithm for Rademacher random multiplicative functions",
  "title_text": "A self-normalized law of the iterated logarithm for Rademacher random multiplicative functions",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:53Z",
  "url": "https://research.draconitic.com/paper/595ngaq4cw3xt",
  "venue": "arXiv",
  "versions": []
}