ON THE UNIVERSALITY OF THE HURWITZ ZETA-FUNCTION

It is known that the Hurwitz zeta-function zeta(s, alpha) with transcendental or rational parameter alpha is universal in the sense that its shifts zeta(s + i tau, alpha), tau is an element of R, approximate with a given accuracy any analytic function uniformly on compact subsets of the strip D = {s is an element of C : 1/2 < sigma < 1}. Let H(D) denote the space of analytic functions on D equipped with the topology of uniform convergence on compacta. In the paper, the classes of functions F : H(D) -> H(D) such that F(zeta(s, alpha)) is universal in the above sense are considered. For example, if F is continuous and, for each polynomial p = p(s), the set F-1 {p} is non-empty, then F(zeta(s, alpha)) with transcendental alpha is universal.