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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.01028 |
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Table of Contents:
- 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 $U(ν,z)$ 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 \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 $H_ν(z)$ is the generalized Hermite polynomial.