{
  "abstract": "Let $f$ be a $1$-bounded multiplicative function, and suppose that $|α-a/q|\\leq 1/(qx)$, where $(a,q)=1$ and $q\\leq x$. We prove that $$ \\left|\\sum_{n\\leq x}\\frac{f(n)}{n}e( nα)\\right| \\leq \\left(\\frac{4}{\\sqrt{6}}+o(1)\\right)\\frac{\\log x}{\\sqrt{q}}+O(\\log q). $$ Here the constant $4/\\sqrt{6}$ is optimal. This settles two conjectures of Granville and Lamzouri recently. If $f$ is completely multiplicative, the constant improves to $1$. The main new ingredient is a two-prime Ramaré decomposition identity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22081",
  "author_total": 1,
  "authors": [
    {
      "name": "Haozhe Gou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7nzxpnem3wqtm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22081",
  "fulltext_url": null,
  "id": "7nzxpnem3wqtm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22081",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7222424045",
  "pdf_url": "https://arxiv.org/pdf/2610.22081",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7nzxpnem3wqtm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.22081",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22081"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Logarithmic exponential sums with multiplicative coefficients",
  "title_text": "Logarithmic exponential sums with multiplicative coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/7nzxpnem3wqtm",
  "venue": "arXiv",
  "versions": []
}