Riemann's auxiliary Function. Basic Results

2024/06/04 by Juan Arias de Reyna, de Reyna, J. Arias
Mathematics · #Algebraic and Geometric Analysis #FOS: Mathematics #History and Overview (math.HO) #Meromorphic and Entire Functions #Number Theory (math.NT) #Primary 11M06 #Secondary 30D10 #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2406.02403

openalex publication_date 2024/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give the definition, main properties and integral expressions of the auxiliary function of Riemann \mathop\mathcal R (s). For example we prove π^-s/2Γ(s/2)\mathop\mathcal R (s)=-\frace^{-πi s/4} s∫_-1^-1+i∞ τ^s/2ϑ₃'(τ) dτ. Many of these results are known, but they serve as a reference. We give the values of \mathop\mathcal R (s) at integers except at odd natural numbers. We have ζ(\tfrac12+it)=e^-iϑ(t)Z(t), \mathop\mathcal R (\tfrac12+it)=\tfrac12e^-iϑ(t)(Z(t)+iY(t)), with ϑ(t), Z(t) and Y(t) real functions.

Related

• Integral Representations of Riemann auxiliary function
• Simple bounds for the auxiliary function of Riemann
• Mean Values of the auxiliary function
• Trivial zeros of Riemann auxiliary function
• Riemann's Auxiliary Function. Right limit of zeros