  In a letter to Weierstrass Riemann asserted that the number $N_0(T)$ of zeros
of $\zeta(s)$ on the critical line to height $T$ is approximately equal to the
total number of zeros to this height $N(T)$. Siegel studied some posthumous
papers of Riemann trying to find a proof of this. He found a function
$\mathop{\mathcal R }(s)$ whose zeros are related to the zeros of the function
$\zeta(s)$. Siegel concluded that Riemann's papers contained no ideas for a
proof of his assertion, connected the position of the zeros of
$\mathop{\mathcal R }(s)$ with the position of the zeros of $\zeta(s)$ and
asked about the position of the zeros of $\mathop{\mathcal R }(s)$. This paper
is a summary of several papers that we will soon upload to arXiv, in which we
try to answer Siegel's question about the position of the zeros of
$\mathop{\mathcal R }(s)$. The articles contain also improvements on Siegel's
results and also other possible ways to prove Riemann's assertion, but without
achieving this goal.
