{
  "abstract": "Let $χ$ be a Dirichlet character mod $D$ with $L(s,χ)$ its associated $L$-function, and let $ψ(x,q,a)$ be, as usual, Chebyshev's prime-counting function for the primes of the arithmetic progression $a$ (mod $q$) with $(a,q)=1$. Let $χ$ be a primitive character modulo $D$, and let $ν>0$ be small. We prove that if $L(s,χ)$ has a Siegel zero at $s=β=1-\\frac{1}{η\\log D}$ with $η>η_0(ν)$ for some large $η_0(ν)$, there exists a range of $x$ for which the asymptotic $$ψ(x,q,a)=\\frac{ψ(x)}{ϕ(q)}\\left[1+O(\\varepsilon_{η_0})-χ\\left(\\frac{aD}{(q,D)}\\right)\\right]$$ holds for $q<x^{\\frac{30}{59}-ν}$. We also show slightly better bounds for $q$ if we take an average over a range of $q$, finding an Elliott-Halberstam-type result for $q\\sim Q$ on the range $Q<x^{\\frac{16}{31}-ν}$. This improves on a 2003 result of Friedlander and Iwaniec that requires $q<x^{\\frac{233}{462}}$ and builds on recent work of Sachpazis.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35950",
  "author_total": 1,
  "authors": [
    {
      "name": "Thomas Wright"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4e80d3axj6n3a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35950",
  "fulltext_url": null,
  "id": "4e80d3axj6n3a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35950",
  "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": "W7214949497",
  "pdf_url": "https://arxiv.org/pdf/2609.35950",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4e80d3axj6n3a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.35950",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35950"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Primes in arithmetic progressions and Siegel zeroes",
  "title_text": "Primes in arithmetic progressions and Siegel zeroes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/4e80d3axj6n3a",
  "venue": "arXiv",
  "versions": []
}