A Standard Method to Prove That the Riemann Zeta Function Equation Has No Non-Trivial Zeros

A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. Suppose ( ) ( ) ( ) 1 2 , , 0 s a b i a b ξ ξ ξ = + = but ( ) ( ) ( ) 1 2 , , 0 s a b i a b ζ ζ ζ = + ≠ with s a ib = + at first. By comparing the real part and the imaginary part of Zeta function equation individually, a set of equation about a and b is obtained. It is proved that this equation set only has the solutions of trivial zeros. In order to obtain possible non-trivial zeros, the only way is to suppose that ( ) 1 , 0 a b ζ = and ( ) 2 , 0 a b ζ = . However, by using the compassion method of infinite series, it is proved that ( ) 1 , 0 a b ζ ≠ and ( ) 2 , 0 a b ζ ≠ . So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.


Introduction
In the author's previous paper titled "The inconsistency problem of Riemann Zeta function equation" [1], it was proved that after complex continuation was considered, on the real axis, the Riemann Zeta function equation had serious inconsistency. The Riemann hypothesis was meaningless [1].
In the present discussions of Riemann [1]. In this paper, a standard method is proposed to separate the real part and the imaginary part of Zeta function equation completely. Then by comparing the real part and the imaginary part individually, it is proved strictly that on whole complex plane, the Zeta function equation has no non-trivial zeros. All trivial zeros are located on the real axis. The Riemann hypothesis is proved untenable again.
The Riemann Zeta function has two forms. One is the form of series summation and another is the form of integral. The form of series summation is more Substituting Equation (1) and Equation (3) in Equation (2), Equation (2) (2) and (4) do not hold.
2) The Riemann Zeta function equation has serious inconsistency. The so-called continuation of function indicates that a function which has no meaning in a certain domain is re-defined so that it becomes meaningful in this domain. But 3) A summation formula was used in the deduction of the integral form of Riemann zeta function. The applicable condition of this formula is 0 x > . At point 0 x = , the formula becomes meaningless. However, the lower limit of Zeta function integral is 0 x = , so this formula can not be used. The integral form of Riemann zeta function does not hold.
of Jacobi function was used to prove the symmetry of Zeta function. The applicable condition of this formula is 0 x > [4]. But the lower limit of integral involved in the deduction is 0 x = . Therefore, the formula can not be used too, the symmetry ( ) ( ) does not hold. The zeros calculation of Riemann Zeta function were discussed in the paper [1]. At present, it has been proved by manual and computer numerical methods that there are lot of zeros on the critical line of 1 2 a = . The number has exceeded 10 trillion [6]. The paper pointed out that all methods used in the calculations were approximate ones. For example, Equation (5) (5) into real and imaginary parts, Equation (5) is written as the forms that real part and imaginary part are separated completely. Then we discuss the zeros of real part and imaginary part individually.
At first, suppose that 1 0 ξ = and 2 0 ξ = , but 1 0 ζ ≠ and 2 0 ζ ≠ , we obtain a set of equation about a and b . It is proved that the only solution to this equation set is 1 a = and 0 b = . But they are the trivial zeros located on the real axis, not non-trivial zeros. So Equation (5) has no non-trivial zeros. By the same method, it also is proved that Equation (4) only has no trivial zeros which are located at the points 2 a n = − ( 0,1,2, n = ) and 0 b = .
At last, in order to obtain possible non-trivial zeros, we take 1 0 ζ = and 2 0 ζ = , i.e., the summation form of Zeta function itself is equal to zero. However, by using the compassion method of infinite series, it is proved that 1 ζ and 2 ζ can not be zeros simultaneously.
Therefore, we prove that the Riemann Zeta function equation has no non-trivial zeros again, the Riemann hypothesis does not hold.

The Proof That the Zeta Function Equation (5) Has No Non-Trivial Zeros
We discuss the zeros of Equation (5) in this section. Then discuss the zeros of Equation (4) in next section.
Here 1 ξ , 2 ξ , 1 ζ and 2 ζ are real functions. By using formula We get ( ) Here 1 G and 2 G are real functions. On the other hand, the definition of real Gama function is [6] ( ) Let a s a ib → = + , we obtain the complex continuation of Gama function.
We have Γ and 2 Γ are also real functions. Therefore, according to the equations above, Equation (5) can be written as The real part and imaginary part of Equation (16) are separated with ( ) ( ) If the Zeta function equation has zeros, its real part and imaginary part should be equal to zero simultaneously. Let 1 0 ξ = and 2 0 ξ = , we obtain ( ) ( ) If 1 0 ζ ≠ and 2 0 ζ ≠ , we can obtain from Equation (19) Substitute Equation (21) in Equation (20), we get Because it is the square summation of two items, each one in Equation (23) should be zero simultaneously To square them and add them together, we get ( ) ( ) Equation (28)  Besides, if we want to look for the non-trivial zeros of ( ) s ξ , the last way is to let 1 0 ζ = and 2 0 ζ = in which the non-trivial zeros may be contained. In this case, the problem whether or not the series summation form of Zeta function can be equal to zero is involved. We will discuss this problem in Section 4. s π π π π π π π π π π π π π π π − − − − + −

The Proof That the Zeta Function Equation (4) Has No Non-Trivial Zeros
Let Here 1 Γ and 2 Γ are also real functions. Thus, Equation (4) can be written By separating real part and imaginary part, we get ( ) ( ) If the Zeta function equation has zeros, its real part and imaginary part should be zeros simultaneously. Suppose that 1 0 ζ ≠ and 2 0 ζ ≠ , according to the same method as shown in Section 2, we get We discuss the zeros of Equations (50) and (51) in next section.

The Convergence of Summation Form of Zeta Function
In order to discuss the zeros of the summation form of Zeta function, we should discuss its convergence. If s a = is a real number, Equation (1) is divergent when 1 a < without zeros. When 1 a > , the series is convergent and great than zero, so (1) has no zeros. The proof is as below [7].
For any 0 x > , we have ( ) Because the radius of convergence is 1, we can not judge the convergence of Equation (1). By using the Euler formula, we write Equation (1)  The radius of convergence is still equal to 1. The convergences of Equation (56) and Equation (57) can not be determined. Because the formulas contain Trigonometric functions, the items in the formula can be positive or negative, we can not ensure that the result of summation is always great than zero. This is different from the situation when s is a real number.

The Zeros of Common Analytic Functions
In the theory of complex functions, the analytic nature of functions is very important. Many theorems cannot be used for non-analytic function. For example, the residue theorem is effective only for analytic functions. On the other hand, a complex function can always be written as Here z x iy = + . If ( ) f z is analytic one, its real part and imaginary part are related. The Cauchy-Riemann formula should be satisfied with [6] , In the current calculation of the zero point of Riemann Zeta function, some approximate methods are adopted. Because Equation (61) is ignored, what obtained are not real zeros of Zeta function [1].
It should be emphasized that when we calculate the zero points of analytic functions, we need to separate the real part and the imaginary part. Because it's possible to have a situation where the real part or the imaginary part is equal to zero, but they are not equal to zero simultaneously. However, in the current zero calculation of Riemann hypothesis, the real part and imaginary part are often mixed together, making the problem ambiguous.

The Proof That the Series Summation Formula of Riemann Zeta Function Has No Zeros on the Complex Plane
It is easy to prove that Equations (62) and (63) satisfy Equation (61). So the summation form of Zeta function is an analytic one. We prove below that Equations (62) and (63) can not be equal to zero simultaneously.
Let's discuss the simplest situation to take the first two items in ( ) However, these three relations are contradictory. So Equations (69) and (70) can not be equal to zero simultaneously.
Off cause, to the series of which item's number is limited, the proof above is not strict. But for the series with infinite items, this method is standard one. Let