Some Problems On L-functions

The Riemann hypothesis,one of the Millennium Problems,has long been paid much attention to for Mathematicians.In 1989,Selberg defined a rather general class of Dirichlet series having an Euler product,analytic continuation and a functional equation of Riemann-type.His aim was to study the value distribution of liner combinations of L-functions.At the same time,he formulated some fundamental conjecture concerning them,and remarked that the conjectures were related to some other classical conjecture in number theory.Since then,this so called Selberg class L-functions became a hot research field of complex analysis and an important object of research in analytic number,yet it is still not understood very well.In fact,Selberg conjectured that the Riemann hypothesis holds for every function in this class.Concerning the proportion of the simple zeros is one of important results which is derived from the Riemann hypothesis.Mathe-maticians generally conjecture that all the non-trivial zeros are simple,this so called the simple zeros hypothesis.Although this proposition has not been proven,there are also many numerical and analytical results as well as the Riemann hypothesis.Steuding showed an asymptotic formula for the c-values of the function L(s),ie.roots of the equation L(s)-c,and gave applications in Nevanlinna theory in[54].Many scholars interest in this research and then do a lot of work concerning this,which connecting two subjects successfully.Recently,Hu and Li established a new function by the zeros which locate at the critical line,and showed that the Riemann hypothesis shall be equivalent to a uniqueness problem of meromorphic functions.In this dissertation,we study the zero distribution and uniqueness problem of L-functions in the extend Selberg class using value distribution theory and other analytic tools.It consists of five parts and the matters are explained as below.In Chapter 1,we introduce the basic knowledge of Selberg class and Nevanlinna theory.In Chapter 2,we investigate the simple zero distribution of the Dirichlet L-functions.We establish some inqualities about the number of distinct zeros of Dirichlet L-functions employing value distribution theory and the abc con-jecture of the functions.In addition,we point out that there exists a positive percentage of the simple zeros of L-a except for at most two values,and give the lower bound of this proportion.In Chapter 3,we discuss the zero distribution of the derivatives of L-functions in the extend Selberg class.We first show that the existence of zero-free regions to the left and right for L(k)(s),and then estimate the number of zeros of L(k)(s).In Chapter 4,we study the uniqueness problems on L-functions in the extend Selberg class sharing sets.The results extend the theorems given by Steuding in[55]and Li in[43].In Chapter 5,we investigate the uniqueness problems with respect to L-functions in the extend Selberg class and meromorphic functions sharing values.The results extend the theorems given by Li in[42]and Garunkstis,Grahl and Steuding in[22].