The Growth Of Dirichlet Series In The Whole Plane

This paper deals with the growth of Dirichlet series of zero order and finiteorder in the whole plane. The full text can be divided into three parts:In the first part, briefly describes the origins and the development of Dirichletseries, and introduces the basic concepts of Dirichlet series. Then, some primaryresults are obtained in this paper.In the second part, introduces the type-functionU (σ)according to the articleof Gao Zongsheng [9], defines the growth on the type-function, by using the methodof Knopp-Kojima, discusses the relation between coefficients and the regular growthof Dirichlet series of zero order and finite order in the whole plane under thecondition σ_a=+∞, and obtains theorem2.1, theorem2.2and theorem2.3.For the third part, introduces the type-functionU (σ)andU ’(σ)according tothe article of Gao Zongsheng [9] and Liu Mingsheng [14], defines the growth on thetype-function, by using the properties of Newton polygon and the method ofKnopp-Kojima,,discusses the relation between coefficients and the growth of lowerorder of Dirichlet series of zero order and finite order in the whole plane under thecondition σ_a=+∞, and obtains theorem3.1,theorem3.2, theorem3.3,theorem3.4and theorem3.5.