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 a collection of analytic functions by shifts ( ζ ( s + i t k α 1 ) , … , ζ ( s + i t k α r ) ) of the Riemann zeta function. Here, { t k : k ∈ N } is the sequence of Gram numbers, and α 1 , … , α r 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.