On Siegel results about the zeros of the auxiliary function of Riemann

2024/06/12 by Juan Arias de Reyna, de Reyna, Juan Arias
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #Analytic and geometric function theory #FOS: Mathematics #Number Theory (math.NT) #Primary 11M06 #Secondary 30D99

paper · pdf · doi:10.48550/arxiv.2406.07968

openalex publication_date 2024/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We state and give complete proof of the results of Siegel about the zeros of the auxiliary function of Riemann \mathop\mathcal R(s). We point out the importance of the determination of the limit to the left of the zeros of \mathop\mathcal R(s) with positive imaginary part, obtaining the term -√(T/2π)P(√(T/2π)) that would explain the periodic behaviour observed with the statistical study of the zeros of \mathop\mathcal R(s). We precise also the connection of the position on the zeros of \mathop\mathcal R(s) with the zeros of ζ(s) in the critical line.

Related

• Report on some papers related to the function \mathop\mathcal R (s) found by Siegel in Riemann's posthumous papers
• Trivial zeros of Riemann auxiliary function
• Asymptotic Expansions of the auxiliary function
• Regions without zeros for the auxiliary function of Riemann
• Infinite Product of the Riemann auxiliary function