We prove that there are at least $1.284 \cdot \sqrt{s/\log s}$ irrational numbers among $\zeta (3), \zeta (5), \zeta (7), \dots , \zeta (s-1)$ for any sufficiently large even integer $s$. This result improves the constant of a previous work. The proof combines the elimination technique of Fischler–Sprang–Zudilin (2019) with the $\Phi _n$ factor method of Zudilin (2001).
Nous prouvons qu’il y a au moins $1.284 \cdot \sqrt{s/\log s}$ nombres irrationnels parmi $\zeta (3), \zeta (5), \zeta (7), \dots , \zeta (s-1)$ pour tout entier pair $s$ suffisamment grand. Ce résultat améliore la constante d’un travail antérieur. La démonstration combine la technique d’élimination de Fischler–Sprang–Zudilin (2019) avec la méthode du facteur $\Phi _n$ de Zudilin (2001).
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Keywords: Irrationality, zeta values, hypergeometric series
Li Lai 1
CC-BY-ND 4.0
Li Lai. A note on the number of irrational odd zeta values II. Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 2, pp. 469-487. doi: 10.5802/jtnb.1368
@article{JTNB_2026__38_2_469_0,
author = {Li Lai},
title = {A note on the number of irrational odd zeta values {II}},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {469--487},
year = {2026},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {38},
number = {2},
doi = {10.5802/jtnb.1368},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1368/}
}
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[1] Irrationalité de $\zeta (2)$ et $\zeta (3)$, Journées Arithmétiques (Luminy, 1978) (Astérisque), Volume 61, Société Mathématique de France, 1979, pp. 11-13 | Zbl | Numdam | MR
[2] Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs, Invent. Math., Volume 146 (2001) no. 1, pp. 193-207 | Zbl | DOI | MR
[3] Linear independence of odd zeta values using Siegel’s lemma, J. Lond. Math. Soc. (2), Volume 113 (2026) no. 4, e70535, 42 pages | Zbl | DOI | MR
[4] Many odd zeta values are irrational, Compos. Math., Volume 155 (2019) no. 5, pp. 938-952 | Zbl | DOI | MR
[5] Small improvements to the Ball–Rivoal theorem and its $p$-adic variant (2024) | arXiv
[6] A note on the number of irrational odd zeta values, Compos. Math., Volume 156 (2020) no. 8, pp. 1699-1717 | Zbl | DOI | MR
[7] At least two of $\zeta (5),\zeta (7),\ldots ,\zeta (35)$ are irrational, Publ. Math. Debr., Volume 101 (2022) no. 3-4, pp. 353-372 | Zbl | DOI | MR
[8] La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs, C. R. Acad. Sci. Paris, Volume 331 (2000) no. 4, pp. 267-270 | Zbl | DOI
[9] Infinitely many odd zeta values are irrational. By elementary means (2018) | arXiv | Zbl
[10] One of the numbers $\zeta (5)$, $\zeta (7)$, $\zeta (9)$, $\zeta (11)$ is irrational, Usp. Mat. Nauk, Volume 56 (2001) no. 4, pp. 149-150 translation in Russ. Math. Surv. 56 (2001), no. 4, p. 774-776 | Zbl
[11] Irrationality of values of the Riemann zeta function, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 66 (2002) no. 3, pp. 49-102 translation in Izv. Math. 66 (2002), no. 3, p. 489-542 | Zbl | MR
[12] Arithmetic of linear forms involving odd zeta values, J. Théor. Nombres Bordeaux, Volume 16 (2004) no. 1, pp. 251-291 | Zbl | Numdam | DOI | MR
[13] One of the odd zeta values from $\zeta (5)$ to $\zeta (25)$ is irrational. By elementary means, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 14 (2018), 028, 8 pages | Zbl
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