Hrybel O. Asymptotic estimates of sums of Dirichlet series.

The object of the study is the entire Dirichlet series and the Dirichlet series absolutely convergent in a half-plane, with a non-negative sequence of exponents increasing to +∞. The aim of the study is to describe the asymptotic properties of the sum of the Dirichlet series in terms of the sequence of its coefficients. Research methods. The dissertation employs modern methods of function theory, along with certain ideas, techniques, and approaches developed by M.M. Sheremeta, P.V. Filevych, and T.Ya. Glova. Main results and scientific novelty. In this dissertation, the following results are obtained for the first time: for each entire (absolutely convergent in a half-plane) Dirichlet series, the associated Dirichlet series is constructed (are constructed), and an approximation theorem is proved that gives an estimate for the sum of the given series in terms of the maximum term of the associated series; relative asymptotic estimates are obtained for the sum of the given series and the maximum term of the associated series, necessary and sufficient conditions on the coefficients of an arbitrary entire (absolutely convergent in a half-plane) Dirichlet series are established to ensure the validity of several general asymptotic estimates for its sum, and also necessary and sufficient conditions on the coefficients of an arbitrary entire (absolutely convergent in a half-plane) Dirichlet series are obtained to ensure the validity of certain global upper estimates for its sum; the necessity of the condition obtained by P. V. Filevych and M. M. Sheremeta on the simultaneous regular variation of the logarithms of the supremum modulus and the maximum term of an entire Dirichlet series is substantiated; a Cauchy–Hadamard-type formula is proved for calculating the generalised order of the logarithm of the maximum term of a Dirichlet series; the growth of Dirichlet series absolutely convergent in a half-plane is investigated in terms of modified orders. Field of application. The dissertation has a theoretical nature. Its results can be applied in further research in the theory of analytic functions and in other areas of mathematics where Dirichlet series or power series arise.