The Growth Of Dirichlet Series And Random Dirichlet Series

In this paper, we first study the growth and regular growth of Dirichlet series of finite order by type function in the plane and obtain two necessary and sufficient conditions; and prove that the growth of random entire functions defined by random Dirichlet series of finite order in every horizontal straight line is almost surely equal to the growth of entire functions defined by their corresponding Dirichlet series.Then we define the hyper-order of Dirichlet series of infinite order respectively in the plane or in the right-half plane,study the relations between the hyper-order and regular hyper-order of Dirichlet series of infinite order and the cofficients;obtain the hyper-order of random entire functions defined by random Dirichlet series of infinite order in every horizontal straight line is almost surely equal to the hyper-order of entire functions defined by their corresponding Dirichlet series.