Mean Values of the auxiliary function

Abstract

Let $\mathcal{R}(s)$ 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 \[\frac{1}{T}\int_0^T |\mathop{\mathcal R}(σ+it)|^2\Bigl(\frac{t}{2π}\Bigr)^σ\,dt, \quad\text{and}\quad \frac{1}{T}\int_0^T |\mathop{\mathcal R}(σ+it)|^2\,dt.\] 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.

arXiv comment9 pages, 1 figure