  Let $\mathop{\mathcal R}(s)$ be the function related to $\zeta(s)$ found by
Siegel in the papers of Riemann. In this paper we obtain the main terms of the
mean values \[\frac{1}{T}\int_0^T |\mathop{\mathcal
R}(\sigma+it)|^2\Bigl(\frac{t}{2\pi}\Bigr)^\sigma\,dt, \quad\text{and}\quad
\frac{1}{T}\int_0^T |\mathop{\mathcal R}(\sigma+it)|^2\,dt.\] Giving complete
proofs of some result of the paper of Siegel about the Riemann Nachlass. Siegel
follows Riemann to obtain these mean values. We have followed a more standard
path, and explain the difficulties we encountered in understanding Siegel's
reasoning.
