Universal approximants of the Riemann zeta-function

Kurzfassung/Abstract

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)–(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 G_δ-set in the set of all functions satisfying (i)–(iii).