The Riemann hypothesis is part of Hilbert’s eighth problem in David Hilbert’s list of 23 unsolved problems. It is also one of the Clay Mathematics Institute’s Millennium Prize Problems. Some mathematicians consider it the most important unresolved problem in pure mathematics. Many mathematicians made a lot of efforts; they don’t have to prove the Riemann hypothesis. In this paper, I use the analytic methods to deny the Riemann Hypothesis; if there’s something wrong, please criticize and correct me.
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”
analytical methods, and refute the Riemann Hypothesis. For convenience, we will abbreviate the Riemann Hypothesis as RH.
In this paper,
Lemma 2.1. If
where Re w is the real part of complex number w.
Let
If
where
See [
Lemma 2.2. If
where
Let s is any complex number, we have
where
We write
where Im s is the imaginary part of complex number s.
See [
Lemma 2.3. Let
where
See [
Lemma 2.4. Assume that RH, If
where
See [
Lemma 3.1. Assume that RH, and
where
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 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 Lemma 2.1 and Lemma 3.2,
By Lemma 2.1, Lemma 3.1 and Lemma 3.2, we have
When
Similarly,
Assume that RH and
same as above
When
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
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
Proof. By Lemma 2.4, we have
This completes the proof of Lemma 3.8.
When
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.
Jinhua Fei, (2016) About the Riemann Hypothesis. Journal of Applied Mathematics and Physics,04,561-570. doi: 10.4236/jamp.2016.43061