Density theorems for Riemann's auxiliary function

Abstract: We prove a density theorem for the auxiliar function $\mathop{\mathcal R}(s)$ found by Siegel in Riemann papers. Let $\alpha$ be a real number with $\frac12< \alpha\le 1$, and let $N(\alpha,T)$ be the number of zeros $\rho=\beta+i\gamma$ of $\mathop{\mathcal R}(s)$ with $1\ge \beta\ge\alpha$ and $0<\gamma\le T$. Then we prove [N(\alpha,T)\ll T^{\frac32-\alpha}(\log T)^3.] Therefore, most of the zeros of $\mathop{\mathcal R}(s)$ are near the critical line or to the left of that line. The imaginary line for $\pi^{-s/2}\Gamma(s/2)\mathop{\mathcal R}(s)$ passing through a zero of $\mathop{\mathcal R}(s)$ near the critical line frequently will cut the critical line, producing two zeros of $\zeta(s)$ in the critical line.