Some Study On Value Distribution Of Two Kinds Of Special Functions

In 1920s, Rolf Nevanlinna introduced the characteristic functions of meromorphic functions and established the famous Nevanlinna theory, which had promoted the develop-ment of the value distribution theory of meromorphic functions. From then on, Nevanlinna theory has been well developed itself and widely applied to the researches of normal family, uniqueness of meromorphic functions, complex differential equations, complex dynamics and so forth. Especially, with the development of cross subjects and the birth of new math-ematical tools, many classic complex analysis problems have been solved, for instance, problem on the Nevanlinna second fundamental theorem concerning small functions. Hay-man problem, Gross problem and so on. As a powerful and useful tool, Nevanlinna theory has been employed in many other fields. In this thesis, we will study the value distribution of L-functions in Selberg class and Airy functions under the guidance of my supervisor. Selberg class is a class of the functions which arc important in analytic number theory for their relationship to the Riemann hypothesis. Airy functions are ordinary and widely used functions in ordinary differential equation and in physics.The thesis is structured as follows.Section one is preliminary knowledge. We present the basic Nevanlinna theory and some classical theory of the uniqueness results.In section two, we mainly discuss the value distribution of L-functions in Selberg class. Selberg class is introduced and some previous results on L-functions sharing values are presented. Then we study the uniqueness of L-functions sharing sets. The results in this paper extend Theorem 1 in Li Baoqin [A result on value distribution of L-functions, Proc. Amer. Math. Soc.,138(6):2071-2077,2010]. Moreover, we show by illustrating numbers of examples that our results are precise in certain senses.In section three, we mainly discuss the value distribution of Airy functions. We present Airy functions and some previous results. Then we research Airy functions by Nevanlinna theory, then obtain some interesting results.In section four, the summary and some problems are presented.