Results On Value Distribution Of L-functions And Algebroid Functions

In 1920s, Finland mathematician Rolf Nevanlinna introduced the value distribution of meromorphic functions, which is one of the greatest achieve-ments in mathematics in the 20th century. The theory is considered to be the basis of modern complex analysis, and it has a very important effect on several complex variables. Especially, it is successfully applied to the research of dynamic system and differential equations. This paper investigates value distribution of L-functions and algebroid functions, and we obtain their own characters as special functions.Firstly, Stueding [5] considered the uniqueness of two L-functions; Bao-qin Li (see [8] [12] [14]) improved the results and considered the uniqueness of meromorphic functions and L-functions. when we considere several sharing values, Xiaomin Li [7] also obtained some results. This paper considers mero-morphic functions of order< 1, and we prove several more ordinary results.Theorem 1 Let f be a meromorphic function of order ≤1 with finitely many poles, then f and an L-function L share complex value b CM if and only if for some value k≠b.Theorem 2 Let f be a meromorphic function of order≤1 with finitely many poles and let lim f(s)s→+∞= 1. If f and an L-function L share finite value b (≠1) CM, then L≡f.Since L-functions in the Selberg class include the Riemann zeta functions ζ, we have the following corollary:Corollary 1 Let f be a meromorphic function of order≤1 with finitely many poles and let lims→+∞ f(s)= 1. If f and the Riemann zeta function C share finite value b (≠1) CM, then f≡ζ.The above theorem is also valid for the function sinz, or equivalently, cosz (= sin(z+π/2)), that is,Theorem 3 Let f be an entire function of order≤1 and let f(0)=k for some value k, then f and sin z share value a (≠0) CM if and only if for k, α∈C C and k≠α.For the uniqueness of the algebroid functions, Ullrich [21], Valiron [22], Eremenko [27] and Yuzan He [28] did a lot of work, and Zongsheng Gao [23] and Yuzan He [25] improved the famous 4v+1-value theorem. This paper considers that algebroid functions of order is finite, and we obtain an at most 3v-value theorem.Theorem 4 Let W and M be v-valued algebroid functions such that X(W) is finite, write W/M=H, and let λ(W)≠λ(H). Assume that W and M share 0 CM, if there exist 4v distinct finite non-zero αj ∈ C (j= 1,2,…,4v) such that and then W≡M.In fact, by following Theorem that we obtained, it is easy to see that if v≥2, the minimum of q is not larger than 3v.Theorem 5 Let W and M be v-valued algebroid functions such that λ(W) is finite, write W/M=H, and let λ(W)≠λ(H). If there exist q distinct finite aj∈C and integers kj(≥j)such that where then W三M.The following corollary will give the best possible of value q；furthermore, let kj→+∞,we can get Corollary 3.Corollary 2 Let W and M be v-valued algebroid functions such that λ(W)is finite,write W/M=H,and let λ(W)≠λ(H).If there exist q distinct finite aj∈c and integers kj(≥j)such that then W三M,where u=3,q=2v+2；v≥4,q=2v+3；v≥13,q=2v+4.Corollary 3 Let W and M be v-valued algebroid functions such that λ(W)is finite,write W/M=H,and let λ(W)≠λ(H). If there exist q distinct finite aj∈C such that then W≡M,where v=3,g=2v+2；v≥4,q=2v+3；v≥13,q=2v+4.The dissertation is structured as follows:in chapter 1,we simply introduce the basic concepts and primary resluts of Nevanlinna theory；in chapter 2,we investigate value distribution of L-functions；in chapter 3,we characterize the uniqueness of algebroid functions,and obtain several better results.