I know that the Riemann Zeta function has an infinite number of zeros on the critical line $\sigma = 1/2$;
that it is possible to determine how many zeros the Riemann Zeta function has on any interval $[T, T+H]$;
that the Riemann Zeta function is defined:
by improper integral $\zeta(s) = s\int_{0}^{\infty}{\frac{[x]-x+\frac{1}{2}}{x^{s+1}}}dx$;
or by other improper integral $\Gamma(s)\zeta(s) = \int_{0}^{\infty}{\frac{x^{s-1}dx}{e^x-1}}$
has the first approximate equation $\zeta(s)=\sum_{n\le x}{\frac{1}{n^s}}-\frac{x^{1-s}}{1-s}+\mathcal{O}(x^{-\sigma})$;
and the second approximate equation $\zeta(s)=\sum_{n\le x}{\frac{1}{n^s}}+\chi(s)\sum_{n\le y}{\frac{1}{n^{1-s}}}+\mathcal{O}(x^{-\sigma})+\mathcal{O}(|t|^{1/2-\sigma}y^{\sigma-1})$;
I know how to calculate the non-trivial zeros of the Riemann Zeta function...
But I can't understand why the Riemann Zeta function has nontrivial zeros and why they lie on the critical line $\sigma = 1/2$?
Many thanks for the answers and the link.
Unfortunately, I was hoping to get a non-trivial answer, not the answer "they are", what they are there Hardy proved in 1914. The main word in my question is "why", i.e. a philosophical question that requires a philosophical answer, i.e. an answer that penetrates into the essence of things. I expected to get something like the following answer: Take the second approximate equation, and equate the $x=y=\Big[\sqrt{\frac{t}{2\pi}}\Big]$; then give the sum of $X_n=\frac{1}{n^s}$, $Y_n=\chi(s)\frac{1}{n^{1-s}}$ and the remainder term $R$ vectors, then for $\sigma=1/2$ we get a mirror-symmetric vector system, which forms a closed polyline at the points of non-trivial zeros. As is known from analytic geometry, the sum of the vectors that form a closed polyline is zero.
For example, root #4525 at $s=0.5+5006.208381106i$
