The Dirichlet Series And Random Dirichlet Series Of Zero Order

This paper deals with the growth of Dirichlet series and random Dirichlet series of zero order in the two sapects:1. The Dirichlet series and random Dirichlet series of zero order in the right half-plane.2. The Dirichlet series and random Dirichlet series of zero order in the whole plane.In the first Part ,we outline the history of the researches on Dirichlet series,and give some primary results in this paper. In the Section 1 of Part 2 ,we reference to the type function of Xiong Qing Lai, introduce the type-function U (r )( r=1/σ), define the order on the type -function,and deal with the growth of Dirichlet series of zero order under the condition and the growth index and obtain theorem 2.1.1 and theorem 2.1.2. In the Section 2 of Part 2,we study the relaion between coefficients and the growth of random Dirichlet series of zero order in the right half-plane. When the random variablesfit {X n(ω)} satisfies the certain condition,we prove the growth of random analytic function defined by random Dirichlet series of zero order in every horizontal straight half-line is almost surely equal to the growth of functions defined by its corresponding Dirichlet series, and obtain theorem 2.2.1 and theorem 2.2.2.In the Section 1 of Part 3 ,under the weaker condition the relaion 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.In the Section 2 of Part 3 ,we discuss the growth of random Dirichlet series of zero order in the whole plane,and obtain theorem 3.2.1 and theorem 3.2.2 and theorem 3.2.3.