Received 1 February 2016; accepted 27 March 2016; published 30 March 2016
1. Introduction
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 “super” 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 the
analytical methods, and refute the Riemann Hypothesis. For convenience, we will abbreviate the Riemann Hypothesis as RH.
2. Some Theorems in the Classic Theory
In this paper, is the Euler gamma function, is the Riemann zeta function.
Lemma 2.1. If, then
where Re w is the real part of complex number w.
Let be given, when and, then
If, then
where if, if.
See [1] page 523, page 525.
Lemma 2.2. If, then
where is the Mangoldt function.
Let s is any complex number, we have
where be the nontrivial zeros of, be the positive constant.
We write If then
where Im s is the imaginary part of complex number s.
See [2] page 4, page 31, page 218.
Lemma 2.3. Let is the number of zeros of in the rectangle then
where
See [3] page 98.
Lemma 2.4. Assume that RH, If, then
where.
See [3] page 113.
3. Some Preparation Work
Lemma 3.1. Assume that RH, and, then
where is the ordinate of nontrivial first zero of,
Proof. By Lemma 2.2 and RH, we have
because
and
therefore
And because
therefore
Similarly, we have
This completes the proof of Lemma 3.1.
Throughout the paper, we write
It is easy to see that
Lemma 3.2. We calculate the three complex numbers.
Because
therefore when t is the real number, we have
the three complex numbers required below.
Lemma 3.3.
Proof. By Lemma 2.1 and Lemma 3.2, we have
This completes the proof of Lemma 3.3.
Lemma 3.4.
Proof. By Lemma 2.1 and Lemma 3.2, we have
we write
This completes the proof of Lemma 3.4.
Lemma 3.5.
Proof. When, by Lemma 2.1, we have
By Lemma 2.1 and Lemma 3.2, we have
This completes the proof of Lemma 3.5.
Lemma 3.6. Assume that RH, then
where
Proof. By Lemma 3.2, it is easy to see that
We write
It is easy to see that
Assume that RH and, by the contour integration method, we have
By Lemma 2.1 and Lemma 3.2,
By Lemma 2.1, Lemma 3.1 and Lemma 3.2, we have
When, we have
Similarly,
Assume that RH and, by the contour integration method, we have
same as above
When, we have
Similarly,
Synthesize the above conclusion, we have
therefore
Similarly,
therefore
Similarly,
Therefore
We use the same process, we can get
This completes the proof of Lemma 3.6.
Lemma 3.7. Assume that RH, we have
where be the ordinates of the nontrivial zeros of.
Proof.
by Lemma 2.3, the above formula
By Lemma 3.4, the above formula
by Lemma 3.5 and Lemma 3.6, above formulas.
By Lemma 2.1 and Lemma 3.2, we have
This completes the proof of Lemma 3.7.
Lemma 3.8. Assume that RH, if, then
Proof. By Lemma 2.4, we have
This completes the proof of Lemma 3.8.
4. Conclusions
When, n is the positive integer; by Lemma 2.1, we have
By Lemma 2.2, we have
By Lemma 2.2 and RH, the above formula is
By Lemma 3.3 and Lemma 3.7, the above formula is
By Lemma 3.8, we get a contradiction; therefore the RH is incorrect.