In the paper, we prove that the set of discrete shifts of the Riemann zeta-function (ζ(s + 2πia1k), . . . , ζ(s + 2πiark)), k ∈ N, approximating analytic nonvanishing functions f1(s), . . . , fr(s) defined on {s ∈ C : 1/2 < Res < 1} has a positive density in the interval [N,N + M] with M = o(N), N → ∞, with real algebraic numbers a1, . . . , ar linearly independent over Q. A similar result is obtained for shifts of certain absolutely convergent Dirichlet series.