TITLE:
A Standard Method to Prove That the Riemann Zeta Function Equation Has No Non-Trivial Zeros
AUTHORS:
Xiaochun Mei
KEYWORDS:
Riemann Hypothesis, Riemann Zeta Function, Riemann Zeta Function Equation, Jacobi’s Function, Residue Theorem, Cauchy-Riemann Equation
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.10 No.2,
February
28,
2020
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.