Computational Methods and Function Theory

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

Universal Approximants of the Riemann Zeta-Function

  • Markus Nieβ


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).


Universality tangential approximation Riemann zeta-function 

2000 MSC

11M06 30E10 


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Copyright information

© Heldermann  Verlag 2009

Authors and Affiliations

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

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