An Integral representation of $\mathop{\mathcal R}(s)$ due to Gabcke
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917709039534080 |
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| author | de Reyna, Juan Arias |
| author_facet | de Reyna, Juan Arias |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01028 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Integral representation of $\mathop{\mathcal R}(s)$ due to Gabcke de Reyna, Juan Arias Number Theory Primary 11M06, Secondary 30D99 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. |
| title | An Integral representation of $\mathop{\mathcal R}(s)$ due to Gabcke |
| topic | Number Theory Primary 11M06, Secondary 30D99 |
| url | https://arxiv.org/abs/2407.01028 |