Zeros of $\mathop{\mathcal R}(s)$ on the fourth quadrant
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917696083329024 |
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| author | de Reyna, Juan Arias |
| author_facet | de Reyna, Juan Arias |
| contents | We show that there is a sequence of zeros of $\mathop{\mathcal R}(s)$ in the fourth quadrant. We show that the $n$-th zero $ρ_{-n}=β_{-n}+iγ_{-n}$, with $β_{-n}\sim 4π^2 n/\log^2n$ and $γ_{-n}\sim-4πn/\log n$. We give the first terms of an asymptotic development of $ρ_{-n}$ and an algorithm to calculate $ρ_{-n}$ from $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_11279 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zeros of $\mathop{\mathcal R}(s)$ on the fourth quadrant de Reyna, Juan Arias Number Theory Primary 11M06, Secondary 30D99 We show that there is a sequence of zeros of $\mathop{\mathcal R}(s)$ in the fourth quadrant. We show that the $n$-th zero $ρ_{-n}=β_{-n}+iγ_{-n}$, with $β_{-n}\sim 4π^2 n/\log^2n$ and $γ_{-n}\sim-4πn/\log n$. We give the first terms of an asymptotic development of $ρ_{-n}$ and an algorithm to calculate $ρ_{-n}$ from $n$. |
| title | Zeros of $\mathop{\mathcal R}(s)$ on the fourth quadrant |
| topic | Number Theory Primary 11M06, Secondary 30D99 |
| url | https://arxiv.org/abs/2406.11279 |