{
  "abstract": "In his paper, ''Sur la distribution des fractions irréductibles de dénominateur $n$ ou de dénominateur au plus égal à $x$'', Hubert Delange considers the family integrals defined by \\begin{equation*} \\int_0^1\\left(\\frac{N(n,t)}{ϕ(n)}-t\\right)^{2q}dt, \\end{equation*} where $n>1$ and $q>0$ are positive integers, $ϕ$ is the Euler totient function, and for any real number $t>0$, $N(n,t)$ counts the number reduced fractions whith denominator $n$ that belong to the interval $]0,t]$. He was able to evaluate the integral for $q=1$ but stated that ''Malheureusement, nous n'avons pas su jusqu'à présent faire cette évaluation pour $q>1$''. In addition to its intrinsic interest, this calculation would help understand the asymptotic distribution of the $ϕ(n)$ totatives of $n$ relative to $ϕ(n)$ equidistributed numbers in $[0,n]$. In this paper, we first provide some general properties of the integral for any $q>1$ and reduce the problem to its calculation for odd square-free integers with at least two distinct prime factors. We then analyse into more detail the case $q=2$ for which we provide a formula that reduces the problem to the evaluation of the fourfold Möbius sum, of integer valued modular sums related to generalized Dedekind sums.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14249",
  "author_total": 1,
  "authors": [
    {
      "name": "Steven Brown"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0gvxt7g4qmba0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14249",
  "fulltext_url": null,
  "id": "0gvxt7g4qmba0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14249",
  "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": "W7221759237",
  "pdf_url": "https://arxiv.org/pdf/2610.14249",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0gvxt7g4qmba0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14249",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14249"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the calculation of an integral introduced by Hubert Delange",
  "title_text": "On the calculation of an integral introduced by Hubert Delange",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/0gvxt7g4qmba0",
  "venue": "arXiv",
  "versions": []
}