On shifts of periodic zeta-function in short intervals

The periodic zeta-function ζ(s; a), s = σ + it, a = ₘ ∈ C : m ∈ N\, in the half-plane σ > 1 is defined by Dirichlet series with periodic coefficients aₘ, and has the meromorphic continuation to the whole complex plane. The function ζ(s; a) is a generalization of the Riemann zeta-function and Dirichlet L-functions. In the paper, using only the periodicity of the sequence a, we obtain that the shifts ζ(s + iτ; a), τ ∈ R, approximate a certain class of analytic functions, defined in the strip ∈ C : 1/2 < σ < 1\. For T23/70 H T1/2, the set of such shifts has a positive lower density in the interval $[T, T + H]$, T → ∞. The case of positive density is also discussed. For the proof, the mean square estimate in short intervals for the Hurwitz zeta-function, and probabilistic limit theorems are applied.