Riemann Hypothesis and Value Distribution Theory

Riemann Hypothesis was posed by Riemann in early 50’s of the 19th century in his thesis titled “The Number of Primes less than a Given Number”. It is one of the unsolved “Supper” problems of mathematics. The Riemann Hypothesis is closely related to the well-known Prime Number Theorem. The Riemann Hypothesis states that all the nontrivial zeros of the zeta-function lie on the “critical line” { } 1 : Re 2 s s = . In this paper, we use Nevanlinna’s Second Main Theorem in the value distribution theory, refute the Riemann Hypothesis. In reference [7], we have already given a proof of refute the Riemann Hypothesis. In this paper, we gave out the second proof, please read the reference.


Introduction
In the 19th century, the famous mathematician E. Picard obtained the pathbreaking result: Any non-constant entire function ( ) f z must take every finite complex value infinitely many times, with at most one exception.Later, E. Borel, by introducing the concept of the order of an entire function, gave the above result a more precise formulation.
This result, generally known as the Picard-Borel theorem, lays the foundation for the theory of value distribution and since then it has been the source of many research papers on this subject.R. Nevanlinna made the decisive contribution to the development of the theory of value distribution.The Picard-Borel Theorem is a direct consequence of Nevanlinna theory.
In this paper, we use Nevanlinna's Second Main Theorem in the value distribu-J.H. Fei tion theory; we got an important the conclusion by Riemann hypothesis.This conclusion contradicts the References [5] theorem 8.12 of the page 204, therefore we prove that Riemann hypothesis is incorrect.

Some Results in the Theory of Value Distribution
We give some notations, definitions and theorems in the theory of value distribution, its contents are in the references [1] and [6].We write log 1 log 0 0 1 , n r f the number of poles of ( ) , each pole being counted with its proper multiplicity.Denote by ( ) n f the multiplicity of the pole of ( ) f z at the origin.For arbitrary complex number , a ≠ ∞ we denote by the number of zeros of ( ) the multiplicity of the zero of ( ) We write ( ) are the zeros and poles of ( ) respectively, each zero or pole appears as its multiplicity indicates, and 0 z = is neither zero nor pole of the function ( ) f z , then, in the circle z ρ < , we have the following formula This formula is called Jensen formula.Lemma 2.2 follows from the References [1], page 3. Lemma 2.3.Let ( ) f z be the meromorphic function in the circle , z R ≤ and ( ) exists, and that ( ) The lemma 2.4 follows from the References [2], the theorem 8.2 of page 87.
where Re s is the real part of the complex number s.

Preparatory Work
Let s it σ = + is the complex number, when 1 σ > , Riemann zeta function is ( ) ( ) Proof.In the Lemma 2.2, we choose ( ) ( ) This completes the proof of Lemma 3.3.

Proof of Conclusion
Theorem.If RH is correct, when   The conclusion of Theorem contradicts the References [5] theorem 8.12 of the page 204, therefore we prove that Riemann hypothesis is incorrect.
2.1 follows from the References [1], page 7. Lemma 2.2.Let ( ) f z be a non-constant meromorphic function in the circle ( ) Riemann zeta function.Lemma 2.5 follows from the References [3], the lemma 8.4 of page 188.Lemma 2.6.Let ( ) f z be the analytic function in the circle , Re f z on z r = re- spectively.Then for 0 r R < < , we have

∑
This completes the proof of Lemma 3.1.Now, we assume that Riemann hypothesis is correct, and abbreviation as RH.In other words, when 1 2 σ > , the function ( ) it ζ σ + has no zeros.The function by Lemma 3.1 and Lemma 3.2, we have 4 1 log log log .
is the analytic function, and it have neither zeros nor poles in the circle z R ≤ , we have

≤
This completes the proof of Theorem.
[1]s is a form of Nevanlinna's Second Main Theorem.Lemma 2.3 follows from the References[1], theorem 2.4 of page 55.