Integral Representations of Riemann auxiliary function
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909238862807040 |
|---|---|
| author | de Reyna, Juan Arias |
| author_facet | de Reyna, Juan Arias |
| contents | We prove that the auxiliary function $\mathop{\mathcal R}(s)$ has the integral representation \[\mathop{\mathcal R}(s)=-\frac{2^s π^{s}e^{πi s/4}}{Γ(s)}\int_0^\infty y^{s}\frac{1-e^{-πy^2+πωy}}{1-e^{2πωy}}\,\frac{dy}{y},\qquad ω=e^{πi/4}, \quad\Re s>0,\] valid for $σ>0$. The function in the integrand $\frac{1-e^{-πy^2+πωy}}{1-e^{2πωy}}$ is entire. Therefore, no residue is added when we move the path of integration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02016 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integral Representations of Riemann auxiliary function de Reyna, Juan Arias Number Theory Primary 11M06, Secondary 30D99 We prove that the auxiliary function $\mathop{\mathcal R}(s)$ has the integral representation \[\mathop{\mathcal R}(s)=-\frac{2^s π^{s}e^{πi s/4}}{Γ(s)}\int_0^\infty y^{s}\frac{1-e^{-πy^2+πωy}}{1-e^{2πωy}}\,\frac{dy}{y},\qquad ω=e^{πi/4}, \quad\Re s>0,\] valid for $σ>0$. The function in the integrand $\frac{1-e^{-πy^2+πωy}}{1-e^{2πωy}}$ is entire. Therefore, no residue is added when we move the path of integration. |
| title | Integral Representations of Riemann auxiliary function |
| topic | Number Theory Primary 11M06, Secondary 30D99 |
| url | https://arxiv.org/abs/2407.02016 |