Mean Values of the auxiliary function

TL;DR

This paper derives the main terms of mean values related to Siegel's auxiliary function connected to the Riemann zeta function, clarifying Siegel's original results with a more standard approach.

Contribution

It provides a detailed derivation of mean value formulas for Siegel's auxiliary function, clarifying and completing Siegel's original results with new proofs.

Findings

Main terms of mean values of the auxiliary function are obtained.

Complete proofs of Siegel's results on mean values are provided.

The approach clarifies the difficulties in understanding Siegel's reasoning.

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.