{
  "abstract": "Let $m\\geq 2$ be an integer, and let $d\\geq 2$ be an even integer. We study monic, reciprocal, non-cyclotomic polynomials $$ P(x)=x^d+a_{d-1}x^{d-1}+\\cdots+a_1x+1\\in\\mathbb{Z}[x]. $$ For a fixed residue class $r\\in\\{0,1,\\ldots,m-1\\}$, we prove that there exists an effectively computable constant $c_{r,m}>1$ such that the Mahler measure $$ M(P)\\geq c_{r,m} $$ for every irreducible polynomial $Q$ whose inner coefficients satisfy $$ a_j\\equiv r\\pmod m \\qquad (1\\leq j\\leq d-1). $$ Moreover, we explicitly compute the constant $c_{r,m}$ in terms of $r$ and $m$. For the residue classes $r=2$, $r=3$, and $r=m-1$, we obtain improved lower bounds for suitable ranges of the modulus~$m$. We also consider monic, reciprocal, irreducible, non-cyclotomic polynomials whose inner coefficients belong to two distinct residue classes modulo $m$. More precisely, if the inner coefficients with odd indices are congruent to $a\\pmod m$ and those with even indices are congruent to $b\\pmod m$, where $a\\not\\equiv b\\pmod m$, then we prove the existence of a constant $c_{m,a,b}>1$ such that the Mahler measure $$ M(P)\\geq c_{m,a,b}. $$ These results solve Lehmer's problem for several families of polynomials. We also construct infinitely many examples in every even degree $d\\geq 2$ of monic, reciprocal, irreducible, non-cyclotomic polynomials over $\\mathbb{Q}$, together with explicit information about their Mahler measures. Our proofs rely on Y. Bilu's equidistribution theorem and its effective version proved by C. D'Andrea \\emph{et al.}, as well as on a method developed by P. Borwein \\emph{et al.} While it is natural to consider Bilu's equidistribution results in conjunction with the Borwein {\\it et al}. method to Lehmer's problem, we are not aware of any previous work in the literature that employs this particular approach.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33631",
  "author_total": 2,
  "authors": [
    {
      "name": "Piyush Kumar Jha"
    },
    {
      "name": "R. Thangadurai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7xyxmcwyvrvh6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33631",
  "fulltext_url": null,
  "id": "7xyxmcwyvrvh6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33631",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7224016329",
  "pdf_url": "https://arxiv.org/pdf/2610.33631",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7xyxmcwyvrvh6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33631",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33631"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Lehmer's Problem",
  "title_text": "On Lehmer's Problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/7xyxmcwyvrvh6",
  "venue": "arXiv",
  "versions": []
}