All of zeros of Riemann's Zeta-Function are on $σ$=1/2

Abstract

The research shows that Riemann proved that all of zeros of Riemann's zeta function are on $σ=1/2$ based on the functional equation \begin{align*} π^{-\frac{s}{2}}Γ\left( \frac{s}{2} \right) ζ(s)&={\frac{1}{s(s-1)} + \int\limits_1^\infty ψ(x) \left( x^{\frac{s}{2} - 1} + x^{-\frac{1+s}{2}} \right) \,dx,}\quad\qquad{s}=σ+it, \end{align*} which is in Riemann's ``Über die Anzahl der Primzahlen unter einer gegebenen Grosse". According to the geometric meaning of the functional equation and the argument principle, we obtain the number of zeros $N_0(T)$ of the Riemann zeta function on the critical segment $σ=1/2,0\leq{t}\leq{T}$ and the number of zeros $N(T)$ of the Riemann zeta function in the rectangular region $-1\leqσ\leq{2},0\leq{t}\leq{T}$, respectively. The result is \begin{align*} N(T)&=N_0(T)=\frac{\arg{\left[π^{-\frac{s}{2}}Γ\left(\frac{s}{2} \right)ζ(s)\right]}}π+1\\ &=\frac{T}{2π}\log\frac{T}{2π}-\frac{T}{2π}+O(\log{T}),\qquad{s=1/2+iT}. \end{align*}