{
  "abstract": "Let $f$ be multiplicative, with $|f(p)|\\le A$ at primes and $\\sum_{n\\le x}|f(n)|^2\\le A^2x$ for every $x\\ge1$. If $|α-a/q|\\le q^{-2}$, $(a,q)=1$, and $3\\le R\\le q\\le N/R$, we prove \\[ \\sum_{n\\le N}f(n)\\operatorname{e}(nα) \\ll_A \\frac{N}{\\log N} +\\frac{N}{\\sqrt R}\\sqrt{\\log\\log(3R)} \\] with effective implied constants. Montgomery and Vaughan proved this with second term $NR^{-1/2}(\\log R)^{3/2}$, and, for $1$-bounded functions, Bachman replaced it by $NR^{-1/2}\\sqrt{\\log R\\log\\log R}$. We remove the factor $\\sqrt{\\log R}$ from Bachman's second term while retaining the original coefficient hypotheses of Montgomery and Vaughan. A more precise estimate records the distance from a rational number. The proof combines the Brun-Titchmarsh inequality on short intervals with maximal Fourier estimates derived from the Carleson-Hunt theorem; the local bounds permit arbitrary prime-dependent prefixes. We also prove sharpness of the square-root displacement dependence.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11070",
  "author_total": 1,
  "authors": [
    {
      "name": "Nicolas Robles"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0qyqfjjs77bfy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11070",
  "fulltext_url": null,
  "id": "0qyqfjjs77bfy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11070",
  "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": "W7212252141",
  "pdf_url": "https://arxiv.org/pdf/2609.11070",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0qyqfjjs77bfy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11070",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11070"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Effective estimates for exponential sums with multiplicative coefficients",
  "title_text": "Effective estimates for exponential sums with multiplicative coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/0qyqfjjs77bfy",
  "venue": "arXiv",
  "versions": []
}