Mean Values of the auxiliary function

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.