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

TL;DR

This paper presents a new, shorter integral representation of the auxiliary Riemann function al R(s), replacing the previous form involving the Mordell integral with a more streamlined expression involving Hermite polynomials.

Contribution

It provides a novel, simplified proof of Gabcke's integral formula for al R(s), avoiding the Mordell integral and expressing it in terms of generalized Hermite polynomials.

Findings

New integral representation of al R(s) using Hermite polynomials.

Simplified proof avoiding Mordell integral.

Enhanced understanding of al R(s) structure.

Abstract

Gabcke proved a new integral expression for the auxiliary Riemann function \[\mathop{\mathcal R}(s)=2^{s/2}\pi^{s/2}e^{\pi i(s-1)/4}\int_{-\frac12\searrow\frac12} \frac{e^{-\pi i u^2/2+\pi i u}}{2i\cos\pi u}U(s-\tfrac12,\sqrt{2\pi}e^{\pi i/4}u)\,du,\] where $U(\nu,z)$ 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 \begin{equation}\mathop{\mathcal R}(s)=-2^s \pi^{s/2}e^{\pi i s/4}\int_{-\infty}^\infty \frac{e^{-\pi x^2}H_{-s}(x\sqrt{\pi})}{1+e^{-2\pi\omega x}}\,dx.\end{equation} where $H_\nu(z)$ is the generalized Hermite polynomial.