The Dirichlet Series And Random Dirichlet Series Of Zero Order

This paper researches the growth of Dirichlet series and Random Dirichlet series of zero order in two aspects:1.Dirichlet series and Random Dirichlet series of zero order in the whole plane.2.Dirichlet series and Random Dirichlet series of zero order in the right half-plane.Part one outlines the history of researches on Dirichlet series and presents some primary results obtained in this paper.Section one of Part two references the article of Gao Zongsheng, introduces the type-function U (x), defines the growth on the type-function, and researches the growth of Dirichlet series of zero order in the whole plane under the exponent condition (?), and obtains theorem 2.1.1 and theorem 2.1.2. Section two of Part 2 studies the relation between coefficients and the growth of Random Dirichlet series of zero order in the whole plane, and proves the growth of random analysis function defined by Random Dirichlet series of zero order in every horizontal straight line is almost surely equal to the growth of function defined by its corresponding Dirichlet series, when the random variables sequence { X n(ω)} satisfies the certain condtion, and obtains theorem 2.1.1 and theorem 2.2.2.In section one of Part three under the condtion (?), the relation between coefficients and the growth of Dirichlet series of zero order are discussed ,and theorem 3.1.1 and theorem 3.1.2 are obtained. The section two of Part three discusses the growth of Random Dirichlet series of zero order in the right half-plane, and obtains theorem 3.2.1 and theorem 3.2.2.