Journal of Applied Mathematics and Physics
Vol.05 No.03(2017), Article ID:75211,7 pages
10.4236/jamp.2017.53062
Riemann Hypothesis and Value Distribution Theory
Jinhua Fei
Chang Ling Company of Electronic Technology, Baoji, China
Copyright © 2017 by author(s) and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: March 4, 2017; Accepted: March 28, 2017; Published: March 31, 2017
ABSTRACT
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” . 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.
Keywords:
Value Distribution Theory, Nevanlinna’s Second Main Theorems, Riemann Hypothesis
1. Introduction
In the 19th century, the famous mathematician E. Picard obtained the pathbreaking result: Any non-constant entire function 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 distribution 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.
2. 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
It is easy to see that .
Let be a non-constant meromorphic function in the circle we denote by the number of poles of on , each pole being counted with its proper multiplicity. Denote by the multiplicity of the pole of at the origin. For arbitrary
complex number we denote by the number of zeros of on , each zero being counted with its proper multiplicity. Denote by the multiplicity of the zero of at the origin.
We write
When
and , is called the characteristic function of .
Lemma 2.1. If is a analytical function in the circle we have
where
Lemma 2.1 follows from the References [1] , page 7.
Lemma 2.2. Let be a non-constant meromorphic function in the circle and are the zeros and poles of in the circle respectively, each zero or pole appears as its multiplicity indicates, and is neither zero nor pole of the function , then, in the circle , 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 be the meromorphic function in the circle and when we have
This is a form of Nevanlinna’s Second Main Theorem.
Lemma 2.3 follows from the References [1] , theorem 2.4 of page 55.
Lemma 2.4. Let be decreasing and non-negative for Then the limit
exists, and that Moreover, if then for , we have
The lemma 2.4 follows from the References [2] , the theorem 8.2 of page 87.
Lemma 2.5. When we have
Where is Riemann zeta function.
Lemma 2.5 follows from the References [3] , the lemma 8.4 of page 188.
Lemma 2.6. Let be the analytic function in the circle let and denote the maxima of and on respectively. Then for , we have
where is the real part of the complex number s.
Lemma 2.6 follows from the References [4] , page 175.
3. Preparatory Work
Let is the complex number, when , Riemann zeta function is
When , we have
where is Mangoldt function.
Lemma 3.1. If t is any real number, we have
1)
2)
3)
4)
Proof.
1)
2)
3)
4)
by Lemma 2.4, we have
where
Therefore
This completes the proof of Lemma 3.1.
Now, we assume that Riemann hypothesis is correct, and abbreviation as RH. In other words, when , the function has no zeros. The function is a multi-valued analytic function in the region we choose the principal branch of the function therefore, if then .
Let is the positive constant.
Lemma 3.2. If RH is correct, when we have
Proof. In Lemma 2.6, we choose
Because is the analytic function in the circle , by Lemma 2.6, in the circle , we have
by Lemma 2.5, we have
by Lemma 3.1, we have
therefore, when we have
This completes the proof of Lemma 3.2.
Lemma 3.3. If RH is correct, when in the circle we have
Proof. In the Lemma 2.2, we choose
are the zeros
of the function in the circle each zero appears as its multiplicity indicates. Because the function has no poles in the circle and is not equal to zero, we have
by Lemma 3.1 and Lemma 3.2, we have
Because is neither zero nor pole of the function we have
This completes the proof of Lemma 3.3.
4. Proof of Conclusion
Theorem. If RH is correct, when we have
Proof. In Lemma 2.3, we choose , by Lemma3.1, we have and Because is the analytic function, and it have neither zeros nor poles in the circle , we have
therefore, by Lemma 3.3, we have
In Lemma 2.1, we choose by the maximal principle, in the circle , we have
Therefore, when we have
This completes the proof of Theorem.
The conclusion of Theorem contradicts the References [5] theorem 8.12 of the page 204, therefore we prove that Riemann hypothesis is incorrect.
Cite this paper
Fei, C.L. (2017) Riemann Hypothesis and Value Distribution Theory. Journal of Applied Mathematics and Physics, 5, 734-740. https://doi.org/10.4236/jamp.2017.53062
References
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