On the extension of the Voronin Universality Theorem for the Riemann Zeta-function

Abstract

In the paper, we prove a general universality theorem for the Riemann zeta-function ζ(s) on approximation of a class of analytic functions  by generalized shifts ζ(s + ia(τ )). Here a(τ ) is a real-valued continuous increasing to +∞ function, uniformly distributed modulo  1 and such that |ζ(σ + ia(τ ) + it)| 2 , for σ > 1/2, has a traditional mean estimate for every t ∈ R. For example, the function t v (τ ), v > 0, where t(τ ) is the Gram function, satisfies the hypotheses of the proved theorem. For the proof, the method of weak convergence of  probabilistic measures in the space of analytic functions is developed.