Joint Approximation by the Riemann and Hurwitz Zeta-Functions in Short Intervals

In this study, the approximation of a pair of analytic functions defined on the strip {s=σ+it∈C:1/2<σ<1} by shifts (ζ(s+iτ),ζ(s+iτ,α)), τ∈R, of the Riemann and Hurwitz zeta-functions with transcendental α in the interval [T,T+H] with T27/82⩽H⩽T1/2 was considered. It was proven that the set of such shifts has a positive density. The main result was an extension of the Mishou theorem proved for the interval [0,T], and the first theorem on the joint mixed universality in short intervals. For proof, the probability approach was applied.