The Dirichlet Series And Random Dirichlet Series Of Infinite Order

This paper deals with the growth of Dirichlet series and random Dirichlet series of infinite order in the two sapects:1. The Dirichlet series and random Dirichlet series of infinite order in the whole plane.2. The Dirichlet series and random Dirichlet series of infinite order in the right half-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)= rρ(r) ,(r= e x),define the order on the type-function,and deal with the growth of Dirichlet series of infinite order under the weaker condition and obtain theorem 2.1.1 and theorem 2.1.2.In the Section 2 of Part2,we study the relaion between coefficients and the growth of random Dirichlet series of infinite order in the whole 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 infinite 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 Part3, under the weaker condition the relaion between coefficients and the growth of Dirichlet series of infinite order are discussed ,and theorem 3.1.1 and theorem 3.1.2 are obtained. In the Section 2 of Part3,we discuss the growth of random Dirichlet series of infinite order in the right half-plane,and obtain theorem 3.2.1 and theorem 3.2.2...