Some Researches On Value Distribution Of L-functions And Fermat Tpye Complex Differential Difference Equations

At the beginning of the 20th century,the Finnish mathematician,R.Nevanlinna,established the value distribution theory for the study of meromorphic function value distribution.The theory not only has a milestone significance in the study history of meromorphic function value distribution,but also becomes an indispensable tool for the study of complex analysis.In this paper,we mainly study the value distribution of L-functions in the extended Selberg class and the solutions of Fermat type differential-difference equations by using the Nevanlinna theory.The structure of the paper is organized as follows:In chapter 1,we briefly introduce the common symbols of Nevanlinna theory,the uniqueness theory of meromorphic functions,the theory of L-functions in Selberg class and the difference analogue of logarithmic derivative lemma.In chapter 2,by using the idea of truncated counting functions,we study the uniqueness of transcendental meromorphic functions and L-functions whose certain nonlinear differential polynomials share one finite nonzero value.The result in this paper extends the theorem given by Li,Liu and Yi.In chapter 3,using Nevanlinna theory,we study the uniqueness of nonconstant L-functions in the extended Selberg class,and prove that there exists a finite set S such that f≡L if E_f（S）=E_L（S）,where the nonconstant meromorphic function f is of finitely many poles in the whole complex plane.This is closely related to a question of Gross.In chapter 4,we discuss transcendental entire solution with finite order of the Fer-mat type differential-difference equation[a₀f（z）+a₁f’（z）]²+[bf（z+η）]²=Q（z）e^α（z）.In chapter 5,we explain the main research contents of this paper,and put forward some problems to be solved.