Mean Values of the auxiliary function

Let $(s)$ 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 $$\frac{1}{T}{\int }_0^T\mathrm{\mid }(\sigma +it){\mathrm{\mid }}^2(\frac{t}{2\pi }{)}^{\sigma }dt,\text{and}\frac{1}{T}{\int }_0^T\mathrm{\mid }(\sigma +it){\mathrm{\mid }}^{2}dt\mathrm{.}$$ 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.