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Computational Methods and Function Theory

, Volume 9, Issue 1, pp 145–159 | Cite as

Universal Approximants of the Riemann Zeta-Function

  • Markus Nieβ
Article

Abstract

The Riemann zeta-function ζ(z) has the following well-known properties, cf. the excellent survey of Steuding [10]:
  1. (i)

    it is holomorphic in the complex plane except for a simple pole at z = 1 with residue 1

     
  2. (ii)

    the symmetry relation \(\zeta(z) = \overline {\zeta(\bar z)}\) holds for z ≠ 1

     
  3. (iii)

    the functional equation ζ(z)Γ(z/2)π−z/2 = ζ(1 − z)Γ((1 − z)/2)π − (1−z)/2 holds

     
  4. (iv)

    it has a universality property due to Voronin [11]. The aim of this paper is to show that arbitrarily close approximations of the Riemann zeta-function which satisfy (i)–(iii) may have a different universal property. Consequently, these approximations do not satisfy the Riemann hypothesis. This extends a result due to Gauthier and Zeron [6].

     

Furthermore, we show that the set of all “Birkhoff-universal” functions satisfying (i)–(iii) is a dense Gδ-set in the set of all functions satisfying (i)–(iii).

Keywords

Universality tangential approximation Riemann zeta-function 

2000 MSC

11M06 30E10 

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References

  1. 1.
    G. D. Birkhoff, Démonstration d’une théorème élémentaire sur les fonctions entières, C. R. Acad. Sci. Paris 189 (1929), 473–475.MATHGoogle Scholar
  2. 2.
    J. B. Conway, Functions of One Complex Variable, Springer, New York, Heidelberg, Berlin, 1973.MATHCrossRefGoogle Scholar
  3. 3.
    D. Gaier, Lectures on Complex Approximation, Birkhäuser, Basel/London/Stuttgart, 1987.MATHCrossRefGoogle Scholar
  4. 4.
    P.M. Gauthier and W. Hengartner, Complex approximation and simultaneous interpolation on closed sets, Can. J. Math. 29 (1977), 701–706.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    P. M. Gauthier and M. R. Pouryayevali, Approximation by meromorphic functions with Mittag-Leffler type constraints, Can. Math. Bull. 44 (2001), 420–428.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    P. M. Gauthier and E. S. Zeron, Small perturbations of the Riemann zeta function and their zeros, Comput. Methods Funct. Theory 4 (2004), 143–150.MathSciNetMATHGoogle Scholar
  7. 7.
    K.-G. Groβe-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. 36 (1999), 345–381.MathSciNetCrossRefGoogle Scholar
  8. 8.
    H. Hamburger, Über die Riemannsche Funktionalgleichung der ζ-Funktion, I, II, Math. Zeitschr. 10, 11 (1921), 240–254, 224–245.MathSciNetCrossRefGoogle Scholar
  9. 9.
    L. D. Pustyl’nikov, Rejection of an analogue of the Riemann hypothesis on zeros for an arbitrarily exact approximation of the zeta function satisfying the same functional equation, Uspekhi Mat. Nauk 58 (2003), 175–176 (in Russian); English translation in: Russ. Math. Surv. 58 (2003), 193–194.MathSciNetCrossRefGoogle Scholar
  10. 10.
    J.Steuding, Value-Distribution of L-Functions, Lecture Notes in Mathematics 1877, Springer, 2007.Google Scholar
  11. 11.
    S. M. Voronin, A theorem on the “universality” of the Riemann zeta-function, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 475–486 (in Russian); English translation in: Math. USSR-Izv. 9 (1975), 443–453.MathSciNetMATHGoogle Scholar

Copyright information

© Heldermann  Verlag 2009

Authors and Affiliations

  1. 1.Mathematisch-Geographische FakultätKatholische Universität Eichstätt-IngolstadtEichstättGermany

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