{
  "abstract": "We prove a folklore conjecture on exponential sums with multiplicative coefficients: if $f$ is multiplicative with $|f(n)|\\le1$, $(a,q)=1$, $|α-a/q|\\le q^{-2}$ and $q\\le N/(\\log N)^2$, then \\[ \\sum_{n\\le N}f(n)\\operatorname{e}(nα)\\ll\\frac N{\\log N}+\\frac N{\\sqrt q}. \\] Complete multiplicativity is not required. We also prove two related conjectured bounds: the refinement in which $q$ is replaced by $q(1+N|α-a/q|)$, when this quantity is at most $N/(\\log N)^2$, and the corresponding bound for the sum with logarithmic weights. Finally, we show that the bound can fail for multiplicative $f$ satisfying only $|f(p)|\\le A$ at primes and $\\sum_{n\\le x}|f(n)|^2\\le A^2x$, for every fixed $A>1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19464",
  "author_total": 1,
  "authors": [
    {
      "name": "Nicolas Robles"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/er4zy9b1fhawx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19464",
  "fulltext_url": null,
  "id": "er4zy9b1fhawx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19464",
  "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": "W7222012148",
  "pdf_url": "https://arxiv.org/pdf/2610.19464",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/er4zy9b1fhawx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19464",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19464"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A folklore conjecture on multiplicative exponential sums",
  "title_text": "A folklore conjecture on multiplicative exponential sums",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/er4zy9b1fhawx",
  "venue": "arXiv",
  "versions": []
}