Joint Approximation of Analytic Functions by Shifts of Hurwitz Zeta-Functions in Short Intervals

For j=1,…,r, αj∈(0,1) are such that the set {log(m+αj):m∈N0,j=1,…,r} is linearly independent over Q. In the paper, we consider simultaneous approximation of a tuple of analytic functions by shifts (ζ(s+iτ,α1),…,ζ(s+iτ,αr)) of the Hurwitz zeta-functions. We prove that the set of these shifts has a positive lower density in the interval [T,T+H] for T23/70⩽H⩽T. This extends the known results obtained for algebraically independent αj, j=1,…,r, and with T27/82⩽H⩽T1/2. Moreover, we obtain a theorem on approximation of analytic functions by a linear combination of derivatives of Hurwitz zeta-functions. From that theorem, it follows that the number of zeros of that linear combination lying in the region {s∈C:1/2<σ1<σ2<1,t∈[T,T+H]} is greater than cH with some c>0. This generalizes and extends the classical Voronin result for the function ζ(s,α) with rational α. For the proofs, a probabilistic approach is used.