Value Distribution Of Riemann Zeta-function And Entire Solutions Of A Special PDE

In1920s, R. Nevanlinna introduced the characteristic functions of mero-morphic functions and gave the famous Nevanlinna theory which is one of the greatest achievements in mathematics in the20th century.This theory is considered to be the basis of modern meromorphic function theory, and it has a very important effect on the development and syncretic of many mathematical branches. Especially, Nevanlinna theory has been used as a very powerful tool and has made the research field more active and popular after it was successfully applied to the research of the global analytic solutions of complex differential equations. Taking advantage of the theory just made by himself, R. Nevanlinnastudied the conditions with which a meromorphic function can be determined and obtained two celebrated uniqueness theorems on meromorphic functions, which are called Nevanlinna'S five-value theorem and four-value theorem. Since then, the research of meromorphic functions began. For over a half century,many foreign and domestic mathematicians have devoted themselves to the research and obtained lots of elegant results on the research of the uniqueness theory In the past two decades, Professor Yi Hongxun did much creative work on the uniqueness theory of mero-morphic functions, and well improved the development of the uniqueness theory.The present thesis involves some results of the author that investigate the uniqueness of Riemann zeta-function and the value distribution problems of a special partial differential equation. The dissertation is structured as follows. In Chapter1, we introduce the beneral background of Nevanlinna Theory, some fundamental results, definitions and some notions.In Chapter2, we study the uniqueness theorems of a meromorphic func-tion and Riemann zeta-function. Using the properties of value distribution of Riemann zeta-function, we improve a result given by Bao Qin Li [8] and give a positive answer to a question asked by Chung Chun Yang [41]. We note that our results are sharp. The main theorems are in the following.Theorem1. Leta,b,c∩C∪{∞} be two distinct. If a meromorphic function f in C and the Riemann zeta-function (?) share a, b CM and c IM except possibly at finitely many points, then f=(?)Theorem2. Let a, b∈C∪{∞} be distinct and nonzero. If a meromorphic function f in C and the Riemann zeta-function (?) share a, b CM and O IM except possibly at the points of a set of order less than1, then f=(?).In Chapter3, we characterize entire solutions of a special linear homoge-neous partial differential equation of the second order prove an analogue of Lidelof-Prinsheim thorem determining order and type of entire functions by using their Taylor coefficients. The main theorems are in the following.Theorem3. The partial differential equation (1) has an entire solution u=f(t,z) on C2if and only if u=f(t,z) has a series expansion such that where Tn(t)=cos(n arccos t) are the Chebyshev polynomials.Theorem4. If f(t,z) is an entire solution of (1) defined by (2) and (3), then Further, if0<λ=ord(f)<∞, then the type σ=typ(f) satisfiesIn Chapter4, we establish Wiman-Valiron theory for entire solutions of the equation (1). The main theorem is in the following.Theorem5. If f(t, z) is an entire solution of (1) defined by (2) and (3), we have Further, if ord(f)<∞, we haveIn Chapter5, we give conditions that entire solutions of partial differential equatioin (1) have no finite Picard exceptional values, and deliver two unique theorems on entire solutions. The main theorems are in the following.Theorem6. If f(t, z) is an entire solution of (1) defined by (2) and (3) such that ord(f)<∞, and satisfies one of the following conditions(i) ord(f)=0or ord(f) is not an integer,(ii) ord(f) is a positive integer λ, and for some integer j≥0then f:C2→C sujective.Theorem7. Let f(t,z), g(t,z) be transcendental entire solutions of (1) such that ord(f)<∞, ord(g)<∞. and where q=max{[ord(f)],[ord(g)]}. If there exists a complex number a≠f(0,0) such that f and g share a counting multiplicity, then we have f=g. Theorem8.Let f(t,z) be a non-constant meromorphic solution of (1)such that ord(f)<∞, and let g(t,z) be a non-constant meromorphci function of finite order on C2.Assume that f and g share0,1,∞CM.Then g must be a Mobius transformation of f.Moreover,one of the following four cases is occurred:(a)g=f;(b)gf=1;(c)gf=f+g(d)there exist a constant b≠1and a polynomial βsuch that...