{
  "abstract": "In 2019 La Bretèche, Stoll and Tennenbaum showed that the ratio of the sum of digits function $s_{q_1}(n)/s_{q_2}(n)$ of two multiplicatively independent bases $q_1$ and $q_2$ is dense in $\\mathbb{Q}^+$. Recently Spiegelhofer proved that in the special case $s_2(n)/s_3(n)=1$ we have infinitely many solutions. Spiegelhofer extended this jointly with Drmota to show that the pair $(s_2(n),s_3(n))$ attains almost every value of $\\mathbb{N}^2$ and hence, in particular, that every rational ratio is attained infinitely many times.\\\\ In this paper we show that, indeed, for any pair of multiplicatively independent bases $p$ and $q$, that the ratio attains every rational number infinitely many times. We also study this problem in the multiplicatively dependent case, hence giving a complete characterisation in the case of 2 bases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.14241",
  "author_total": 1,
  "authors": [
    {
      "name": "Pascal Jelinek"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/djqzr2vrwh6g5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.14241",
  "fulltext_url": null,
  "id": "djqzr2vrwh6g5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.14241",
  "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": "W7203610824",
  "pdf_url": "https://arxiv.org/pdf/2608.14241",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/djqzr2vrwh6g5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.14241",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.14241"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Ratio of sum of digits functions in two bases",
  "title_text": "Ratio of sum of digits functions in two bases",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/djqzr2vrwh6g5",
  "venue": "arXiv",
  "versions": []
}