Advances in Pure Mathematics

Volume 10, Issue 2 (February 2020)

ISSN Print: 2160-0368   ISSN Online: 2160-0384

Google-based Impact Factor: 0.62  Citations  h5-index & Ranking

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

HTML  XML Download Download as PDF (Size: 336KB)  PP. 86-99  
DOI: 10.4236/apm.2020.102006    372 Downloads   921 Views  
Author(s)

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.

Share and Cite:

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.

Cited by

No relevant information.

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.