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

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 ) = ξ 1 ( a , b ) + i ξ 2 ( a , b ) = 0 but ζ ( s ) = ζ 1 ( a , b ) + i ζ 2 ( a , b ) ≠ 0 with s = a + i b 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 ζ 1 ( a , b ) = 0 and ζ 2 ( a , b ) = 0. However, by using the compassion method of infinite series, it is proved that ζ 1 ( a , b ) ≠ 0 and ζ 2 ( a , b ) ≠ 0. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.