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: $0 < |W(s)| \neq 1$, $\zeta(s)$ 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: $0 \leq \sigma \leq 1$, but $\sigma \neq 1/2$, 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.