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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