{
  "abstract": "In this short note, we prove that: If $ m \\in \\mathbb{N} $ and $ p_i \\in \\mathcal{P}_0 = \\left\\lbrace p_0=2, p_1 = 3, p_2 = 5, ... \\right\\rbrace $, the set of all positive prime numbers, then for all real numbers $ x \\geq 2, $ the double inequality $$ \\prod_{n=0}^{m-1}\\left(\\frac{p_n^{x}}{p_n^{x}-1}\\right)\\times \\frac{p_m^{x}}{(p_m^{x}-\\lambda)} < \\zeta(x) < \\prod_{n=0}^{m-1} \\left(\\frac{p_n^{x}}{p_n^{x}-1}\\right)\\times \\frac{p_m^{x}}{(p_m^{x}-\\mu)}, $$ where $ \\zeta $ is the Riemann zeta function, holds with the best possible constants $ \\lambda = 1 $ and $ \\mu = p_m^2 \\left[ 1 - \\frac{6}{\\pi^2} \\prod_{n=0}^{m-1} \\frac{p_n^2}{(p_n^2 - 1)} \\right].$ This solves our recently published conjecture and gives several sharp bounds for the even-indexed Bernoulli numbers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Yogesh J Bagul"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1z8pz1jv7smjp.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "1z8pz1jv7smjp",
  "issns": [],
  "landing_url": "https://hal.science/hal-05757913v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05757913/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1z8pz1jv7smjp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:37Z",
      "native_id": "hal-05757913",
      "source": "hal",
      "url": "https://hal.science/hal-05757913v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Inequalities involving the Riemann zeta function and implications for Bernoulli numbers",
  "title_text": "Inequalities involving the Riemann zeta function and implications for Bernoulli numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:37Z",
  "url": "https://research.draconitic.com/paper/1z8pz1jv7smjp",
  "venue": "HAL",
  "versions": []
}