Mean Values of the auxiliary function

2024/06/19 by de Reyna, Juan Arias
#FOS: Mathematics #Number Theory (math.NT) #Primary 11M06 #Secondary 30D99

paper · doi:10.48550/arxiv.2406.13278

Abstract

Let \mathop\mathcal 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 |\mathop\mathcal R(σ+it)|²((t)/(2π))^σ dt, \quadand (1)/(T)∫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.

Related

• Riemann's auxiliary Function. Basic Results
• On Siegel results about the zeros of the auxiliary function of Riemann
• Some mean value results related to Hardy's function
• Simple bounds for the auxiliary function of Riemann
• Mean values of ratios of the Riemann zeta function