  The research shows that Riemann proved that all of zeros of Riemann's zeta
function are on $\sigma=1/2$ based on the functional equation \begin{align*}
  \pi^{-\frac{s}{2}}\Gamma \left( \frac{s}{2} \right)
\zeta(s)&={\frac{1}{s(s-1)} + \int\limits_1^\infty \psi(x) \left(
  x^{\frac{s}{2} - 1} + x^{-\frac{1+s}{2}}
  \right) \,dx,}\quad\qquad{s}=\sigma+it, \end{align*} which is in Riemann's
``\"{U}ber 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 $\sigma=1/2,0\leq{t}\leq{T}$ and the number of zeros $N(T)$ of
the Riemann zeta function in the rectangular region
$-1\leq\sigma\leq{2},0\leq{t}\leq{T}$, respectively. The result is
\begin{align*}
N(T)&=N_0(T)=\frac{\arg{\left[\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}
\right)\zeta(s)\right]}}{\pi}+1\\
&=\frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+O(\log{T}),\qquad{s=1/2+iT}.
\end{align*}
