The Growth Of Dirichlet Series And Random Dirichlet Series Of Finite Order

This paper researches the growth of Dirichlet series and Random Dirichlet series of finite order in two aspects:1.Dirichlet series and Random Dirichlet series of finite order in the right half-plane.2.On the rearrangement of the coefficients of Dirichlet series of finite order.Part one introduced the Dirichlet series and the origins and development of the Random Dirichlet series ,and presents some primary results obtained in this paper. Section one of part two references the article of Niu Yingchun, introduces the type-function U (r ) = rρ( r), researche the growth of Dirichlet series of finite order in the right half-plane under the exponent condition n = loglogn.and improves some known results , 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 finite order in the right half-plane,and obtains when the random variable sequence { X n(ω)} satisfies lemma 2.2.1 and lemma 2.2.2, in the right half-plane, growth of random analysis function which is determined by the Random Dirichlet series is almost surely same with corresponding Dirichlet series in any right half straight line. and obtains theorem 2.2.1 and theorem 2.2.2.Part three deals with the growth of Dirichlet series of finite order after rearrangement of the coefficients. First,the part investigates the Dirichlet series of finite order defined in the right half-plane, under the exponent condition , obtains the sufficient and necessary unvaried condition of the following proximate order after rearranging the coefficients,and obtains theorem 3.1.1.Second, the part discusses the Dirichlet series of finite order defined in the whole plane, when exponent satisfies the certain condition and obtains the unvaried condition of proximate order after rearranging coefficients,and obtains theorem 3.2.1.