{
  "abstract": "Let $λ$ be the Liouville function. We prove Let $λ$ be the Liouville function. We prove \\[ \\mathcal{I}_λ(X)=\\int_{\\mathbb{T}}\\left|\\sum_{1\\leqslant n\\leqslant X}λ\\left(n\\right)\\mathrm{e}\\left(nα\\right)\\right| \\mathrm{d}α \\gg X^{\\frac{1}{4}}. \\] This removes the $X^\\varepsilon$ loss in the previous result of Pandey and Radziwiłł~\\cite{PandeyRadziwill2026}. In the second paper in this series, we will prove that \\[ \\limsup_{X \\to \\infty} \\frac{\\log \\mathcal{I}_λ(X)} {\\log X} \\geqslant\\frac{1}{3}. \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03589",
  "author_total": 3,
  "authors": [
    {
      "name": "Jiangpeng Li"
    },
    {
      "name": "Yu-Chen Sun"
    },
    {
      "name": "Hui Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fdkj23mfhp5p2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03589",
  "fulltext_url": null,
  "id": "fdkj23mfhp5p2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03589",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03589",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fdkj23mfhp5p2.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.03589",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03589"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the lower bound for the $L^1$ norm of Liouville exponential sums. I",
  "title_text": "On the lower bound for the L^1 norm of Liouville exponential sums. I",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/fdkj23mfhp5p2",
  "venue": "arXiv",
  "versions": []
}