Density theorems for Riemann's auxiliary function

We prove a density theorem for the auxiliar function $(s)$ found by Siegel in Riemann papers. Let $\alpha$ be a real number with $\frac{1}{2}<\alpha \le 1$, and let $N(\alpha ,T)$ be the number of zeros $\rho =\beta +i\gamma$ of $(s)$ with $1\ge \beta \ge \alpha$ and $0<\gamma \le T$. Then we prove $$N(\alpha ,T)\ll T^{\frac{3}{2}-\alpha }(\log T{)}^3\mathrm{.}$$ Therefore, most of the zeros of $(s)$ are near the critical line or to the left of that line. The imaginary line for ${\pi }^{-s\mathrm{/}2}\mathrm{\Gamma }(s\mathrm{/}2)(s)$ passing through a zero of $(s)$ near the critical line frequently will cut the critical line, producing two zeros of $\zeta (s)$ in the critical line.