On Universal Relatives of the Riemann Zeta-function

The Riemann zeta-function zeta has the following well-known properties: (M) It is meromorphic in C with a simple pole at z = 1 with residue 1. (SR) The symmetry relation zeta(z) = <(zeta<(z)over bar>)over bar> holds for z not equal 1. (FE) The following functinal equation holds: zeta(z)Gamma(z/2)pi(-z/2) = zeta(1 - z)Gamma((1 - z)/2)pi(-(1-z)/2). Moreover, zeta has a universality property due to Voronin (1975). We show that arbitrarily close approximations of the Riemann zeta-function that satisfy (M),(SR),(FE) may have a different universality property. Consequently, these approximations do not satisfy the Riemann hypothesis. Moreover, we investigate the set of all "Birkhoff-universal" functions satisfying (M),(SR),(FE).