A Standard Method to Prove That the Riemann Zeta Function Equation Has No Non-Trivial Zeros
Open Access
- 1 January 2020
- journal article
- Published by Scientific Research Publishing, Inc. in Advances in Pure Mathematics
- Vol. 10 (02), 86-99
- https://doi.org/10.4236/apm.2020.102006
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(1-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.Keywords
This publication has 1 reference indexed in Scilit:
- Algebraic Number TheoryPublished by Springer Science and Business Media LLC ,1999