Abstract
In this paper we generalize Nesterenko’s criterion to the case where the small linear forms have an oscillating behaviour (for instance given by the saddle point method). This criterion provides both a lower bound for the dimension of the vector space spanned over the rationals by a family of real numbers and a measure of simultaneous approximation to these numbers (namely, an upper bound for the irrationality exponent if 1 and only one other number are involved). As an application, we prove an explicit measure of simultaneous approximation to ζ(5), ζ(7), ζ(9), and ζ(11), using Zudilin’s proof that at least one of these numbers is irrational.
Similar content being viewed by others
References
Adamczewski, Sur l’exposant de densité des nombres algébriques, International Math. Research Notices, article ID 024, (2007) 6 pages.
R. Apéry, Irrationalité de ζ(2) et ζ(3), Journées Arithmétiques (Luminy, 1978), 61 in Astérisque, (1979), 11–13.
Ball K., Rivoal T.: Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs. Invent. Math. 146, 193–207 (2001)
E. Bedulev, On the linear independence of numbers over number fields, Mat. Zametki [Math. Notes] 64, (1998), 506–517 [440–449].
N. Fel’dman and Y. Nesterenko, Number Theory IV, Transcendental Numbers, Encyclopaedia of Mathematical Sciences 44, Springer, 1998, A.N. Parshin and I.R. Shafarevich, eds.
Fischler S.: Restricted rational approximation and Apéry-type constructions. Indagationes Mathem. 20, 201–215 (2009)
S.Fischler, Nesterenko’s linear independence criterion for vectors, (2011) preprint.
S. Fischler and T. Rivoal, Un exposant de densité en approximation rationnelle, International Math. Research Notices (2006), article ID 95418, 48 pages.
Fischler S., Rivoal T.: Irrationality exponent and rational approximations with prescribed growth. Proc. Amer. Math. Soc. 138, 799–808 (2010)
Fischler S., Zudilin W.: A refinement of Nesterenko’s linear independence criterion with applications to zeta values. Math. Ann. 347, 739–763 (2010)
L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Pure and Applied Mathematics, Wiley, 1974.
Y. Nesterenko, On the linear independence of numbers, Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] 40, (1985), 46–49 [69–74].
Rivoal T.: La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs. C. R. Acad. Sci. Paris, Ser I 331, 267–270 (2000)
Rivoal T.: Irrationalité d’au moins un des neuf nombres ζ(5), ζ(7), . . . , ζ(21). Acta Arith. 103, 157–167 (2002)
V. Sorokin, On the Zudilin-Rivoal Theorem, Mat. Zametki [Math. Notes] 81, (2007), 912–923 [817–826].
Töpfer T.: Über lineare Unabhängigkeit in algebraischen Zahlkörpern. Results Math. 25, 139–152 (1994)
W. Zudilin, One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational, Uspekhi Mat. Nauk [Russian Math. Surveys] 56, (2001), 149–150 [774–776].
W. Zudilin, Irrationality of values of the Riemann zeta function, Izvestiya Ross. Akad. Nauk Ser. Mat. [Izv. Math.]. 66, (2002), 49–102 [489–542].
Zudilin W.: Arithmetic of linear forms involving odd zeta values. J. Théor. Nombres Bordeaux 16, 251–291 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fischler, S. Nesterenko’s criterion when the small linear forms oscillate. Arch. Math. 98, 143–151 (2012). https://doi.org/10.1007/s00013-011-0354-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-011-0354-y