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

TL;DR

This paper discusses a proof that all non-trivial zeros of Riemann's zeta function lie on the critical line =1/2, based on the functional equation and geometric interpretation, confirming the Riemann Hypothesis.

Contribution

It provides a proof that all zeros of Riemann's zeta function are on =1/2 using the functional equation and argument principle, linking geometric meaning to zero distribution.

Findings

Number of zeros on =1/2 matches the total zeros in the critical strip.

Derived the zero counting function N(T) with asymptotic form.

Confirmed all zeros lie on the critical line =1/2.

Abstract

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*}