On simultaneous Diophantine approximations to \(\zeta(2)\) and \(\zeta(3)\). (English) Zbl 1315.11062

The authors construct simultaneous rational approximations to both \(\zeta(2)\) and \(\zeta(3)\) using hypergeometric tools. They prove that if \(\eta>0\) and \(\varepsilon>0\) are given, and \(m\) is sufficiently large with respect to \(\varepsilon\) and \(\eta\), then \(|a_0+ a_1\zeta(2)|> e^{-(s_0+ \eta)m}\) with \(s_0= 6.770732145\dots\), where \((a_0,a_1,a_3)\in \mathbb{Q}^3\setminus\{0\}\) satisfies certain strong divisibility conditions, and \(|a_0|,|a_1|,|a_2|\leq e^{-(\tau_0+\varepsilon)m}\) with \(\tau_0= 0.899668635\dots\). This result implies the irrationality of both \(\zeta(2)\) and \(\zeta(3)\), does not give however the expected linear independence of 1, \(\zeta(2)\) and \(\zeta(3)\).
Moreover, the authors further introduce a new notion of simultaneous Diophantine exponent, and give the basic properties of this new concept, and compare it with the classical irrationality exponent and S. Fischler’s \(\psi\)-exponent of irrationality [Indag. Math., New Ser. 20, No. 2, 201–215 (2009; Zbl 1198.11057)].


11J82 Measures of irrationality and of transcendence
11J72 Irrationality; linear independence over a field
33C20 Generalized hypergeometric series, \({}_pF_q\)


Zbl 1198.11057
Full Text: DOI arXiv


[1] Apéry, Roger, Irrationalité de \(\zeta(2)\) et \(\zeta(3)\), Astérisque, 61, 11-13, (1979) · Zbl 0401.10049
[2] Bailey, D. H.; Borwein, D.; Borwein, J. M.; Crandall, R. E., Hypergeometric forms for Ising-class integrals, Experiment. Math., 16, 3, 257-276, (2007) · Zbl 1134.33016
[3] Ball, Keith; Rivoal, Tanguy, Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs, Invent. Math., 146, 1, 193-207, (2001) · Zbl 1058.11051
[4] Dauguet, Simon, Généralisations du critère d’indépendance linéaire de nesterenko, (2014), Université Paris-Sud, PhD thesis · Zbl 1387.11040
[5] Dauguet, Simon, Généralisations quantitatives du critère d’indépendance linéaire de nesterenko, (2014), submitted for publication · Zbl 1387.11040
[6] Fischler, Stéphane, Restricted rational approximation and apéry-type constructions, Indag. Math., 20, 2, 201-215, (2009) · Zbl 1198.11057
[7] Hata, Masayoshi, A note on beukers’ integral, J. Austral. Math. Soc. Ser. A, 58, 2, 143-153, (1995) · Zbl 0830.11026
[8] Hata, Masayoshi, A new irrationality measure for \(\zeta(3)\), Acta Arith., 92, 1, 47-57, (2000) · Zbl 0955.11023
[9] Rhin, Georges; Viola, Carlo, On a permutation group related to \(\zeta(2)\), Acta Arith., 77, 1, 23-56, (1996) · Zbl 0864.11037
[10] Rivoal, Tanguy, Irrationalité d’au moins un des neuf nombres \(\zeta(5), \zeta(7), \ldots, \zeta(21)\), Acta Arith., 103, 2, 157-167, (2002) · Zbl 1015.11033
[11] Slater, Lucy Joan, Generalized hypergeometric functions, (1966), Cambridge University Press Cambridge
[12] Zudilin, Wadim, One of the numbers \(\zeta(5)\), \(\zeta(7)\), \(\zeta(9)\), \(\zeta(11)\) is irrational, Russian Math. Surveys, 56, 4, 774-776, (2001) · Zbl 1047.11072
[13] Zudilin, Wadim, Arithmetic of linear forms involving odd zeta values, J. Théor. Nombres Bordeaux, 16, 1, 251-291, (2004) · Zbl 1156.11327
[14] Zudilin, Wadim, Approximations to -, di- and tri-logarithms, J. Comput. Appl. Math., 202, 2, 450-459, (2007) · Zbl 1220.65028
[15] Zudilin, Wadim, Arithmetic hypergeometric series, Russian Math. Surveys, 66, 2, 369-420, (2011) · Zbl 1225.33008
[16] Zudilin, Wadim, Two hypergeometric tales and a new irrationality measure of \(\zeta(2)\), Ann. Math. Québec, 38, (2014) · Zbl 1307.11085
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.