Integral Representations of Riemann auxiliary function

2024/07/02 by Juan Arias de Reyna, de Reyna, Juan Arias
Mathematics · #Algebraic and Geometric Analysis #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #Primary 11M06 #Secondary 30D99 #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2407.02016

openalex publication_date 2024/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the auxiliary function \mathop\mathcal R(s) has the integral representation \mathop\mathcal R(s)=-\frac2^s π^se^{πi s/4}Γ(s)∫₀^∞ y^s\frac1-e^-πy2+πωy1-e^2πωy (dy)/(y), ω=e^πi/4, \Re sgt;0, valid for σ>0. The function in the integrand \frac1-e^-πy2+πωy1-e^2πωy is entire. Therefore, no residue is added when we move the path of integration.

Related

• Riemann's auxiliary Function. Basic Results
• An Integral representation of \mathop\mathcal R(s) due to Gabcke
• Mean Values of the auxiliary function
• Integral Representation for Riemann-Siegel Z(t) function
• Trivial zeros of Riemann auxiliary function