Universal Approximants of the Riemann Zeta-Function

The Riemann zeta-function ζ(z) has the following well-known properties, cf. the excellent survey of Steuding [10]: it is holomorphic in the complex plane except for a simple pole at z = 1 with residue 1the symmetry relation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\zeta(z) = \overline {\zeta(\bar z)}$\end{document} holds for z ≠ 1the functional equation ζ(z)Γ(z/2)π−z/2 = ζ(1 − z)Γ((1 − z)/2)π − (1−z)/2 holdsit 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].