An Integral representation of \mathop\mathcal R(s) due to Gabcke

2024/07/01 by Juan Arias de Reyna, de Reyna, Juan Arias
Mathematics · #Advanced Mathematical Identities #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #Primary 11M06 #Secondary 30D99

paper · pdf · doi:10.48550/arxiv.2407.01028

openalex publication_date 2024/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Gabcke proved a new integral expression for the auxiliary Riemann function \mathop\mathcal R(s)=2^s/2π^s/2e^πi(s-1)/4∫_{-\frac12\searrow\frac12} \frace^{-πi u2/2+πi u}2icosπuU(s-\tfrac12,√(2π)e^πi/4u) du, where U(ν,z) is the usual parabolic cylinder function. We give a new, shorter proof, which avoids the use of the Mordell integral. And we write it in the form \mathop\mathcal R(s)=-2^s π^s/2e^{πi s/4}∫_-∞^∞ \frace^-πx2H_-s(x√π)1+e^-2πωx dx. where H_ν(z) is the generalized Hermite polynomial.

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