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Computational Methods and Function Theory 9 (2009), No. 1, 145--159
Copyright Heldermann Verlag 2009

Universal Approximants of the Riemann Zeta-Function

Markus Nieß
markus.niess@ku-eichstaett.de , Katholische Universität Eichstätt-Ingolstadt, Mathematisch-Geographische Fakultät, 85071 Eichstätt, Germany.

[Abstract-pdf] [Abstract-ps]

The Riemann zeta-function ζ(z) has the following well-known properties, cf. the excellent survey of Steuding [10]:
  • (i) it is holomorphic in the complex plane except for a simple pole at z = 1 with residue 1;
  • (ii) the symmetry relation ζ(z) = \overline{ζ(\bar z) holds for z ≠1;
  • (iii) the functional equation ζ(z) Γ(z/2) π-z/2 = ζ(1 - z) Γ((1 - z)/2) π-(1 - z)/2 holds;
  • (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)–(iv) 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)&ndash,(iii).

[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.
[10] J. Steuding, Value-Distribution of L-Functions, Lecture Notes in Mathematics 1877, Springer, 2007.
[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.

Keywords: Universality, tangential approximation, Riemann zeta-function.

MSC 2000: 11M06, 30E10.

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