Author(s)

Xiaochun Mei

Department of Theoretical Physics and Pure Mathematics, Institute of Innovative Physics in Fuzhou, Fuzhou, China.

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.

Riemann Hypothesis, Riemann Zeta Function, Riemann Zeta Function Equation, Jacobi’s Function, Residue Theorem, Cauchy-Riemann Equation

Mei, X. (2020) A Standard Method to Prove That the Riemann Zeta Function Equation Has No Non-Trivial Zeros. Advances in Pure Mathematics, 10, 86-99. doi: 10.4236/apm.2020.102006.