Joint Discrete Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function Twisted by Gram Points

Let θ ( t ) denote the increment of the argument of the product π − s / 2 Γ ( s / 2 ) along the segment connecting the points s = 1 / 2 and s = 1 / 2 + i t , and t n denote the solution of the equation θ ( t ) = ( n − 1 ) π , n = 0 , 1 , … . The numbers t n are called the Gram points. In this paper, we consider the approximation of a collection of analytic functions by shifts in the Riemann zeta-function ( ζ ( s + i t k α 1 ) , … , ζ ( s + i t k α r ) ) , k = 0 , 1 , … , where α 1 , … , α r are different positive numbers not exceeding 1. We prove that the set of such shifts approximating a given collection of analytic functions has a positive lower density. For the proof, a discrete limit theorem on weak convergence of probability measures in the space of analytic functions is applied.