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.