| Abstract [eng] |
The dissertation is devoted to the approximation problems of analytic functions by shifts of some zeta-functions. The results obtained extend the well-known Voronin universality theorem on the approximation of non-vanishing analytic functions defined in the strip D = {s = σ + it ∈ C: ½ < σ < 1} by shifts of the Riemann zeta-function ζ(s). Chapter 1 is devoted to the introduction. Actuality, aims and problems are discussed as well as the history of the problem and main results of dissertation. In Chapter 2, a discrete universality theorem for zeta-functions of normalized Hecke-eigen cusp forms ζ(s, F) on approximation of analytic functions by shifts ζ(s + ihγk, F), h > 0, is obtained. Here {γk: k ∈ N} is a sequence of imaginary parts of non-trivial zeros of the function ζ(s). For this, a modification of the Montgomery pair correlation conjecture is applied. Chapter 3 of the dissertation is devoted to the joint approximation of analytic functions by shifts of Hurwitz zeta-functions with arbitrary parameters. It is proved that there exists a closed non-empty subset of r-dimensional space on analytic functions on D which functions are approximated by shifts (ζ(s + iτ, α1), ... , ζ(s + iτ, αr)) of Hurwitz zeta-functions. In Chapter 4, discrete versions of theorems of Chapter 3 are given. In the last, Chapter 5, a generalization of the Hurwitz zeta-function, the periodic Hurwitz zeta-function, is considered. For the periodic Hurwitz zeta-function with arbitrary parameter and periodic sequence, approximation theorems of a class of analytic functions on D are proved. Moreover, these theorems are generalized for some compositions of the periodic Hurwitz zeta-functions. All above mentioned approximation theorems are proved in terms of a positive lower density and of positive density of the set of approximating shifts. |
