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

In this paper, a theorem is obtained on the approximation in short intervals of collection analytic functions by shifts (ζ(s+itkα1),…,ζ(s+itkαr)) Riemann zeta function. Here, {tk:k∈N} sequence Gram numbers, and α1,…,αr are different positive numbers not exceeding 1. It proved that above set interval [N,N+M], here M=o(N) as N→∞, has lower density. For proof, joint limit for weakly convergent probability measures applied.