Mean Values of the auxiliary function

Let R(s) be the function related to ζ(s) found by Siegel in the papers of Riemann. In this paper we obtain the main terms of the mean values \[1/T∫_0^T |{ R}(σ+it)|^2(t/2π)^σ\,dt, 1/T∫_0^T |{ R}(σ+it)|^2\,dt.\] 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.