Abstract
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 ξ(s) = ξ1(a,b) + iξ2(a,b) = 0 but ζ(s) = ζ1(a,b) + iζ2(a,b) ≠ 0 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 ζ1(a,b) = 0 and ζ2(a,b) = 0. However, by using the compassion method of infinite series, it is proved that ζ1(a,b) ≠ 0 and ζ2(a,b) ≠ 0. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.
Highlights
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
A simple and standard method is proposed to prove that the Riemann Zeta function equation has no non-trivial zeros on whole complex plane
The Riemann hypothesis claims that all non-trivial zeros were located on the critical line Re(s) = 1 2 of complex plane, but it can not be proved up to now
Summary
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. It is proved that on the real axis, the Riemann Zeta function equation only holds at the point s= a= 1 2. A simple and standard method is proposed to prove that the Riemann Zeta function equation has no non-trivial zeros on whole complex plane. We prove that the Riemann Zeta function equation has no non-trivial zeros again, the Riemann hypothesis does not hold. The problem whether or not the series summation form of Zeta function can be equal to zero is involved If the real part and imaginary part of Zeta function ζ (1 − s) are not equal to zeros, the Zeta function equation Equation (4) has no non-trivial zeros. 4. The Proof That the Series Summation Formula of Riemann Zeta Function Has No Zeros
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