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

Abstract

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