Joint Discrete Approximation by Shifts of Hurwitz Zeta-Function: The Case of Short Intervals

Since 1975, it has been known that the Hurwitz zeta-function has a unique property to approximate by its shifts all analytic functions defined in the strip D={s=σ+it:1/2<σ<1}. However, such an approximation causes efficiency problems, and applying short intervals is one of the measures to make that approximation more effective. In this paper, we consider the simultaneous approximation of a tuple of analytic functions in the strip D by discrete shifts (ζ(s+ikh1,α1),…,ζ(s+ikhr,αr)) with positive h1,…,hr of Hurwitz zeta-functions in the interval [N,N+M] with M=max1⩽j⩽rhj−1(Nhj)23/70. Two cases are considered: 1° the set {(hjlog(m+αj),m∈N0,j=1,…,r),2π} is linearly independent over Q; and 2° a general case, where αj and hj are arbitrary. In case 1°, we obtain that the set of approximating shifts has a positive lower density (and density) for every tuple of analytic functions. In case 2°, the set of approximated functions forms a certain closed set. For the proof, an approach based on new limit theorems on weakly convergent probability measures in the space of analytic functions in short intervals is applied. The power η=23/70 comes from a new mean square estimate for the Hurwitz zeta-function.