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

Abstract

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

arXiv comment5 pages 2 figures