Title: Mean Values of the auxiliary function Show Chinese title

Title: 辅助函数的均值

Abstract: 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. Show Chinese abstract

Abstract: 让$\mathop{\mathcal R}(s)$成为与$\zeta(s)$相关的函数，该函数由Siegel在黎曼的论文中找到。 在本文中，我们得到了均值\[\frac{1}{T}\int_0^T |\mathop{\mathcal R}(\sigma+it)|^2\Bigl(\frac{t}{2\pi}\Bigr)^\sigma\,dt, \quad\text{and}\quad \frac{1}{T}\int_0^T |\mathop{\mathcal R}(\sigma+it)|^2\,dt.\]的主要项，给出了对Siegel关于黎曼手稿的一些结果的完整证明。 Siegel遵循黎曼的方法来获得这些均值。 我们遵循了一条更标准的路径，并解释了我们在理解Siegel的推理时遇到的困难。