  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.
