{
  "abstract": "Let $m\\geq 1$ be a fixed integer, $a$ an integer satisfying $(a,m)=1$, and $z\\geq 1$ a real parameter. Denote by $ω_{z}(n;m,a)$ the number of distinct prime divisors $p$ of $n$ satisfying $p\\equiv a\\, (m)$ and $p\\leq z$. We study an asymptotic behaviour of $\\sum_{n\\leq x}\\left(ω_{z}(n;m,a)-\\frac{1}{φ(m)}\\log\\log z\\right)^{k}$ as $x\\to\\infty$ for a wide range of positive integer $k\\geq 2$, where $φ(\\cdot)$ is the Euler function. Following a method of Granville and Soundararajan we lead an asymptotic formula for the above. Also, we investigate $\\sum_{n\\leq x}\\left(ω(n;m,a)-\\frac{1}{φ(m)}\\log\\log x\\right)^{k}$, where $ω(n;m,a)$ denotes the number of distinct prime divisors $p$ of $n$ such that $p\\equiv a\\, (m)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22784",
  "author_total": 3,
  "authors": [
    {
      "name": "T. Makoto Minamide"
    },
    {
      "name": "Haruka Sakai"
    },
    {
      "name": "Yoshio Tanigawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d8qn4d7y716az.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22784",
  "fulltext_url": null,
  "id": "d8qn4d7y716az",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22784",
  "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": "W7214010047",
  "pdf_url": "https://arxiv.org/pdf/2609.22784",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d8qn4d7y716az.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22784",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22784"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On a Turán's theorem for arithmetic progressions",
  "title_text": "On a Turán's theorem for arithmetic progressions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/d8qn4d7y716az",
  "venue": "arXiv",
  "versions": []
}