{
  "abstract": "Let $μ$ be the Möbius function and $e(t)=e^{2πit}$. We prove that if $N\\ge2$, $α\\in\\mathbb{R}$, $(a,q)=1$, and $|α-a/q|\\le q^{-2}$, then \\[\\bigg|\\sum_{n\\le N}μ^2(n)e(αn)\\bigg|\\ll\\left(\\frac Nq+q\\right)(\\log 2N)^5, \\] with an absolute implied constant, and we deduce the corresponding estimate on the minor arcs of the Hardy--Littlewood dissection throughout the range $Q\\le N^{1/2}$. The estimates of Schlage-Puchta [SP] and of Tolev [T] have the same dependence on $q$ and $Q$ but carry a factor $N^{\\varepsilon}$. The proof uses Heath-Brown's square sieve with sieving primes confined to an interval $(P,2P]$, where $P$ may be as small as a multiple of $\\log N$; a finite Fejér majorant in place of a truncated Fourier series; and, after completion of the character sums, a count of representations that exploits the restriction on the primes in place of the divisor function.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03961",
  "author_total": 3,
  "authors": [
    {
      "name": "Nicolas Robles"
    },
    {
      "name": "Alexandru Zaharescu"
    },
    {
      "name": "Dirk Zeindler"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/240r8b4dxdw19.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03961",
  "fulltext_url": null,
  "id": "240r8b4dxdw19",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03961",
  "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": "W7208779579",
  "pdf_url": "https://arxiv.org/pdf/2609.03961",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/240r8b4dxdw19.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03961",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03961"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the exponential sum over squarefree integers",
  "title_text": "On the exponential sum over squarefree integers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/240r8b4dxdw19",
  "venue": "arXiv",
  "versions": []
}