On Universal Relatives of the Riemann Zeta-function

摘要

The Riemann zeta-function ζ 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 ζ(z) - ζ(z) holds for z ≠ 1. (FE) The following functional equation holds:rnζ(z)Γ(z/2)π~(-z/2)=ζ(1-z)Γ((1-z)/2)π~(-(1-z))/2.rnMoreover, ζ 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).