| Abstract [eng] |
In the thesis, the Atkinson formula for the periodic zeta-function on the critical line and the critical strip, and limit theorems in the sense of weak convergence of probability measures in various spaces are considered. The aim of the thesis is to solve the following problems: 1. To obtain the Atkinson formula on the critical line for the periodic zeta-function. 2. To obtain the Atkinson formula in the critical strip for the periodic zeta-function. 3. To prove limit theorems on the complex plane in the sense of weak convergence for the periodic zeta-function. 4. To prove limit theorems in the space of analytic functions for the periodic zeta-function. To solve them analytical and probabilistic methods are applied. For the proof of Atkinson formula, we use properties of the error term in the Dirichlet divisor problem and classical Voronoi formula. For the proof of limit theorems, the theory of weak convergence of probability measures, in particular, the Prokhorov's theory is applied. All results obtained in the thesis are new. The Atkinson formula for periodic zeta-function was not known. The same is true for limit theorems for periodic zeta-function. The Atkinson formula gives the explicit formula for the error term in the asymptotic formula for the first moment. This result is not only interesting itself but also has a series of applications, for example, in the investigation of higher moments. Probabilistic limit theorems are used for the characterization of asymptotic behavior of zeta-functions. Recently, it was observed that theorems of such a kind are the principal component in the proof of universality of zeta-functions. Finally, the periodic zeta-function is not classical, however, it occurs in various problems of analytic number theory. For example, it appears in the asymptotic formula for the mean square with respect to the the Hurwitz and Lerch zeta-functions. On the other hand, the majority works of the mentioned above authors are devoted to the classical zeta-functions, and the results for the periodic function are not numerous. |
