| Dirichlet series was first studied in analytic number theory.On the one hand,it is a generalization of Taylor series,on the other hand,it is also a special case of Laplace-Stieltjes transformation.The growth of Dirichlet series is not only a key element of Dirichlet series,but also the basis of value distribution theory.The sum of convergent Dirichlet series,as a holomorphic function,plays an important role in complex analysis and has certain theoretical value in the study of the growth of Dirichlet series.Based on the research of real exponential Dirichlet series,in this thesis the growth of Dirichet series are studied,by introducing a family of real functions,using the Knopp-Kojima method and constructing the Newton polygon,and some related results of Dirichet series with real exponential are generalized.The specific contents are as follows.In the first chapter,the research background and research situation of Dirichet series are briefly introduced,then some notations and definitions related to Dirichet series are introduced.In the second chapter,the generalized order of Dirichlet series in the half plane and the whole plane are studied.Firstly,by using the Knopp-Kojima method,the relations of the maximum modulus,the maximum term and the coefficients of Dirichlet series are obtained.Secondly,under certain conditions,the above relations are transformed into the relations of the generalized order and its coefficients of Dirichlet series,and some classical results are proved under a weaker condition.In the third chapter,we study the generalized lower order of Dirichlet series in the half plane and the whole plane.For the lower order of Dirichlet series,some coefficient conditions or exponential conditions are required in most classical results.In this chapter,we remove these constraints by the Knopp-Kojima method.Then,the Newton polygon is constructed by the coefficients and exponents of the series,and the relationships between the generalized lower order and its coefficients of Dirichlet series are studied.Thus,some corresponding conclusions on the generalized order of Dirichlet series areobtained. |
