The Growth Of DIRICHLET Series With Finite Positive Order In The Plane

There are two parts in this dissertation, which focuses on two contents: one is the growth of Dirichlet series with finite positive order in the plane, the other is (p, q)(R)-order and (p, q)(R)-type of entire function defined through B-value Dirichlet series.The outline of the paper is arranged as follows:The purpose of part 1 is devoted to study the relationship between coefficients and growth of Dirichlet series with finite positive order in the plane by using Newton polynomial and type function. We get theorems 1.3.1, 1.3.2, 1.3.3, 1.3.4, 1.3.5 and 1.3.6. The results are supplements of Dirichlet series with finite positive order in the plane and some methods are different from zero order and infinite order.In part 2, we try investigating (p, q)(R)-order and (p, q)(R)-type of entire function defined by Dirichlet series in the complex Banach space in first time on the basis of Newton polynomial. We transform the (p, q)(R)-order and type of B-valued Dirichlet series into the research for the (p, q)(R)-order and type of Dirichlet series. Using geometric method and combining the corresponding results mentioned in the references, we obtain the corroding results of (p, q)(R)-order and type of B-valued Dirichlet series. Namely, theorems 2.3.1, 2.3.2, 2.3.3 and 1.3.4. At the same time, we give the Lemmas 1.2.2, 1.2.3, 1.2.4, 1.2.5, 1.2.6 and 2.2.1, 2.2.3,which satisfy the need of theorems'proof.