A Proof of Riemann Hypothesis

Abstract

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