Note on the positivity of the real part of the log-derivative of the Riemann $ξ$-function near the critical line
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917242998882304 |
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| author | Grigutis, Andrius Turčinskas, Lukas |
| author_facet | Grigutis, Andrius Turčinskas, Lukas |
| contents | In this work, we investigate the positivity of the real part of the log-derivative of the Riemann $ξ$-function in the region $1/2+1/\sqrt{\log t}<σ<1$, where $t$ is sufficiently large. We provide an explicit lower bound for $\mathfrak{R}\sum_ρ1/(s-ρ)$, where the summation runs over the zeta-zeros on the critical line. We also consider hypothetical cases of positivity of the log-derivative of the Riemann $ξ$-function in the provided region, assuming that there are non-trivial zeta-zeros off the critical line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_18963 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Note on the positivity of the real part of the log-derivative of the Riemann $ξ$-function near the critical line Grigutis, Andrius Turčinskas, Lukas Number Theory In this work, we investigate the positivity of the real part of the log-derivative of the Riemann $ξ$-function in the region $1/2+1/\sqrt{\log t}<σ<1$, where $t$ is sufficiently large. We provide an explicit lower bound for $\mathfrak{R}\sum_ρ1/(s-ρ)$, where the summation runs over the zeta-zeros on the critical line. We also consider hypothetical cases of positivity of the log-derivative of the Riemann $ξ$-function in the provided region, assuming that there are non-trivial zeta-zeros off the critical line. |
| title | Note on the positivity of the real part of the log-derivative of the Riemann $ξ$-function near the critical line |
| topic | Number Theory |
| url | https://arxiv.org/abs/2509.18963 |