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A Proof of Riemann Hypothesis

General Mathematics 2020-05-05 v2

Abstract

The meromorphic function W(s)W(s) introduced in the Riemann-Zeta function ζ(s)=W(s)ζ(1s)\zeta(s) = W(s) \zeta(1-s) maps the line of s=1/2+its = 1/2 + it onto the unit circle in WW-space. W(s)=0|W(s)| = 0 gives the trivial zeroes of the Riemann-Zeta function ζ(s)\zeta(s). In the range: 0<W(s)10 < |W(s)| \neq 1, ζ(s)\zeta(s) does not have nontrivial zeroes. W(s)=1|W(s)|=1 is the necessary condition for the nontrivial zeros of the Riemann-Zeta function. Writing s=σ+its = \sigma + it, in the range: 0σ10 \leq \sigma \leq 1, but σ1/2\sigma \neq 1/2, even if W(s)=1|W(s)|=1, the Riemann-Zeta function ζ(s)\zeta(s) is non-zero. Based on these arguments, the nontrivial zeros of the Riemann-Zeta function ζ(s)\zeta(s) can only be on the s=1/2+its = 1/2 + it critical line. Therefore a proof of the Riemann Hypothesis is presented.

Keywords

Cite

@article{arxiv.1909.10313,
  title  = {A Proof of Riemann Hypothesis},
  author = {Tao Liu and Juhao Wu},
  journal= {arXiv preprint arXiv:1909.10313},
  year   = {2020}
}

Comments

16 pages, 4 figures

R2 v1 2026-06-23T11:23:07.599Z