Non universality on the critical line

Abstract

We prove that the Riemann zeta-function is not universal on the critical line by using the fact that the Hardy Z-function is real, and some elementary considerations. This is a related to a recent result of Garunkstis and Steuding. We also prove conditional and partial results for non universality on the lines Re(s)=\sigma for 0<\sigma<1/2 and together with our recent result for non universality on the line Re(s)=1 it will mostly answer the question of on what lines the zeta-function is universal.