The periodic Hurwitz zeta function ζ(s, α; A), s = σ + it, 0 < α ≤ 1, is defined, for σ > 1, by ζ(s, α; A) = ∞ m=0 a m /(m + α) s and by analytic continuation elsewhere. Here {a m } is a periodic sequence of complex numbers. In this paper, a discrete universality theorem for the function ζ(s, α; A) with a transcendental parameter α is proved. Roughly speaking, this means that every analytic function can be approximated uniformly on compact sets by shifts ζ(s + imh, α; A), where m is a non-negative integer and h is a fixed positive number such that exp{2π/ h} is rational.

In the paper, a joint universality theorem for the Riemann zeta-function and a collection of periodic Hurwitz zeta-functions on approximation of analytic functions is obtained.

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The periodic zeta-function ζ(s; a), s = σ + it is defined for σ > 1 by the Dirichlet series with periodic coefficients and is meromorphically continued to the whole complex plane. It is known that the function ζ(s; a), for some sequences a of coefficients, is universal in the sense that its shifts ζ(s + iτ ; a), τ ∈ R, approximate a wide class of analytic functions. In the paper, a weighted universality theorem for the function ζ(s; a) is obtained.

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