A Proof of Riemann Hypothesis

Abstract

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

16 pages, 4 figures