On the number of zeros of $\mathop{\mathcal R}(s)$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917692599959552 |
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| author | de Reyna, Juan Arias |
| author_facet | de Reyna, Juan Arias |
| contents | We prove that the number of zeros $\varrho=β+iγ$ of $\mathop{\mathcal R}(s)$ with $0<γ\le T$ is given by \[N(T)=\frac{T}{4π}\log\frac{T}{2π}-\frac{T}{4π}-\frac12\sqrt{\frac{T}{2π}}+O(T^{2/5}\log^2 T).\] Here $\mathop{\mathcal R}(s)$ is the function that Siegel found in Riemann's papers. Siegel related the zeros of $\mathop{\mathcal R}(s)$ to the zeros of Riemann's zeta function. Our result on $N(T)$ improves the result of Siegel. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08890 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the number of zeros of $\mathop{\mathcal R}(s)$ de Reyna, Juan Arias Number Theory Primary 11M06, Secondary 30D99 We prove that the number of zeros $\varrho=β+iγ$ of $\mathop{\mathcal R}(s)$ with $0<γ\le T$ is given by \[N(T)=\frac{T}{4π}\log\frac{T}{2π}-\frac{T}{4π}-\frac12\sqrt{\frac{T}{2π}}+O(T^{2/5}\log^2 T).\] Here $\mathop{\mathcal R}(s)$ is the function that Siegel found in Riemann's papers. Siegel related the zeros of $\mathop{\mathcal R}(s)$ to the zeros of Riemann's zeta function. Our result on $N(T)$ improves the result of Siegel. |
| title | On the number of zeros of $\mathop{\mathcal R}(s)$ |
| topic | Number Theory Primary 11M06, Secondary 30D99 |
| url | https://arxiv.org/abs/2406.08890 |