{
  "abstract": "Let $f:\\mathbb{N}\\to{z\\in\\mathbb{C}:|z|\\leq1}$ be multiplicative, and suppose that $α$ has a rational approximation $a/q$ with $|α-a/q|\\leq 1/(qx)$ and $q\\leq x/((\\log x)^2(\\log\\log x)^3)$. We prove that $\\left|\\sum_{n\\leq x}f(n)e(nα)\\right|\\leq \\left(\\frac4{\\sqrt6}+o(1)\\right)\\frac{x}{\\sqrt q} +O\\left(\\frac{x}{\\log x}\\right)$, whereas complete multiplicativity improves $4/\\sqrt6$ to $1$. Both constants are optimal. In particular, this confirms a folklore conjecture on exponential sums with multiplicative coefficients motivated by the work of Montgomery and Vaughan (1977). We also obtain logarithmically weighted estimates and give counterexamples to related conjectures.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22345",
  "author_total": 1,
  "authors": [
    {
      "name": "Haozhe Gou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fpfkv08fjyqcc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22345",
  "fulltext_url": null,
  "id": "fpfkv08fjyqcc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22345",
  "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": "W7222624163",
  "pdf_url": "https://arxiv.org/pdf/2610.22345",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fpfkv08fjyqcc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.22345",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22345"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On exponential sums with multiplicative coefficients",
  "title_text": "On exponential sums with multiplicative coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/fpfkv08fjyqcc",
  "venue": "arXiv",
  "versions": []
}