{
  "abstract": "Erdős, Odlyzko and Sárközy conjectured that every reduced residue class modulo $q$ contains a product of two primes not exceeding $q$. O. Klurman, A. P. Mangerel and J. Teräväinen (2023) have shown that three primes suffice when $q$ is sufficiently smooth. We study the moduli $q=p^γ$ that are powers of a fixed prime $p$ and we show that the third prime can be taken subpolynomially small. For any $q_\\bullet\\le q$ for which $q_\\bullet\\ge\\exp\\bigl(C(\\log q)^{2/3}(\\log\\log q)^{4/3}\\bigr)$, we give an asymptotic formula for the weighted number of representations of each reduced class as a product of two primes not exceeding $q$ and a third not exceeding~$q_\\bullet$. We also show that the conjecture holds for all but a proportion $\\exp\\bigl(-c(\\log q)^{1/3}(\\log\\log q)^{-1/3}\\bigr)$ of the reduced classes. Here $C$ and $c$ are positive constants that depend only on $p$. Both results are unconditional and they rely on a wide zero-free region with no exceptional zeros for the Dirichlet $L$-functions attached to characters modulo $p^γ$.We also present conditional versions of these results for arbitrary $q$ under a quasi-Riemann hypothesis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19597",
  "author_total": 2,
  "authors": [
    {
      "name": "William Banks"
    },
    {
      "name": "Igor E. Shparlinski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9wbhst6djqhft.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19597",
  "fulltext_url": null,
  "id": "9wbhst6djqhft",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19597",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7222355458",
  "pdf_url": "https://arxiv.org/pdf/2610.19597",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9wbhst6djqhft.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19597",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19597"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the Erdős--Odlyzko--Sárközy conjecture for prime-power moduli",
  "title_text": "On the Erdős--Odlyzko--Sárközy conjecture for prime-power moduli",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/9wbhst6djqhft",
  "venue": "arXiv",
  "versions": []
}