{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:31:54Z","timestamp":1760059914109,"version":"build-2065373602"},"reference-count":6,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T00:00:00Z","timestamp":1753142400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We prove that for the odd integers n\u2208{5,7,9,\u2026,25}, the Riemann zeta value \u03b6(n) is not a Liouville number. Our method applies a general strategy pioneered by Wadim Zudilin and D.V. Vasilyev. Specifically, we construct families of high-dimensional integrals that expand into rational linear combinations of odd zeta values, eliminate lower-order terms to isolate \u03b6(n), and apply Nesterenko\u2019s linear independence criterion. We verify the required asymptotic growth and decay conditions for each odd n\u226425, establishing that \u03bc(\u03b6(n))&lt;\u221e, and thus that \u03b6(n)\u2209L. This is the first unified proof covering all odd zeta values up to \u03b6(25) and highlights the structural barriers to extending the method beyond this point. We also give rigorous upper bounds on \u03bc(\u03b6(n)) for all odd integers n\u2208{5,7,\u2026,25}, using multiple integral constructions due to Vasilyev and Zudilin, elimination of lower zeta terms, and the quantitative version of Nesterenko\u2019s criterion.<\/jats:p>","DOI":"10.3390\/axioms14080546","type":"journal-article","created":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T08:45:18Z","timestamp":1753173918000},"page":"546","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Each \u03b6(n), 5 \u2264 n \u2264 25, Is Not a Liouville Number"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0361-576X","authenticated-orcid":false,"given":"Sidney A.","family":"Morris","sequence":"first","affiliation":[{"name":"School of Engineering, IT and Physical Sciences, Federation University Australia, P.O. Box 663, Ballarat, VIC 3353, Australia"},{"name":"Department of Mathematical and Physical Sciences, La Trobe University, Melbourne, VIC 3086, Australia"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,22]]},"reference":[{"key":"ref_1","first-page":"11","article-title":"Irrationalit\u00e9 de \u03b6(2) et \u03b6(3)","volume":"61","year":"1979","journal-title":"Ast\u00e9risque"},{"key":"ref_2","first-page":"267","article-title":"La fonction z\u00eata de Riemann prend une infinit\u00e9 de valeurs irrationnelles aux entiers impairs","volume":"331","author":"Rivoal","year":"2000","journal-title":"Comptes Rendus Acad. Sci. Paris S\u00e9r. I"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"774","DOI":"10.1070\/RM2001v056n04ABEH000427","article-title":"One of the numbers \u03b6(5), \u03b6(7), \u03b6(9), \u03b6(11) is irrational","volume":"56","author":"Zudilin","year":"2001","journal-title":"Russ. Math. Surv."},{"key":"ref_4","first-page":"693","article-title":"Multiple integrals and the linear forms in odd zeta values","volume":"75","author":"Vasilyev","year":"2004","journal-title":"Mat. Zametki"},{"key":"ref_5","first-page":"46","article-title":"On the linear independence of numbers","volume":"1","author":"Nesterenko","year":"1996","journal-title":"Vestnik Mosk. Univ. Ser. I Mat. Mekh."},{"key":"ref_6","first-page":"883","article-title":"M\u00e9moires et communications","volume":"18","author":"Liouville","year":"1844","journal-title":"Comptes Rendus L\u2019Acad\u00c9mie Sci."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/546\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:13:36Z","timestamp":1760033616000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/546"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,22]]},"references-count":6,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["axioms14080546"],"URL":"https:\/\/doi.org\/10.3390\/axioms14080546","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,7,22]]}}}