The universality of an absolutely convergent series on short intervals

Under consideration is a Dirichlet series depending on a parameter and absolutely convergent in the right half of the critical strip. We prove that the set of shifts of the series approximating a prescribed analytic function without zeros has positive density on the intervals of type $[T, T+H]$, where $T^{1/3}(\log T)^{26/15}\leq H\leq T$, and give this density explicitly.