Abstract
In the paper, we obtain universality theorems for compositions of some classes of operators in multidimensional space of analytic functions with a collection of periodic zeta-functions. The used shifts of periodic zeta-functions involve the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function.
Highlights
Let s = σ + it be a complex variable, and let Analytic Functions by Shifts of ∞Certain Compositions
For the proof of universality, the support of limit measures in limit theorems in the space of analytic functions plays a crucial role: it defines the class of approximated functions
By the Linnik-Ibragimov conjecture, see for example [10], all functions defined in some half-plane by Dirichlet series and having a natural growth of their analytic continuation are universal in the sense of approximating of analytic functions
Summary
The periodic zeta-function ζ (s; a) is defined, for σ > 1, by the Dirichlet series am ζ (s; a) = ∑ s , m m =1 and has analytic continuation to the whole complex plane, except for a simple pole at the point s = 1. This follows from the representation ζ (s; a) =. The aim of this paper is universality theorems for compositions of collections of periodic zeta-functions studied in [22]. The sequence {γk } satisfying estimate (1) was used for the first time in the theory of universality in [27] in the case of the Riemann zeta-function.
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