On a positivity property of the real part of logarithmic derivative of the Riemann $ξ$-function
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916429396180992 |
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| author | Goldštein, Edvinas Grigutis, Andrius |
| author_facet | Goldštein, Edvinas Grigutis, Andrius |
| contents | In this paper we investigate the positivity property of the real part of logarithmic derivative of the Riemann $ξ$-function for $1/2<σ<1$ and sufficiently large $t$. We give an explicit upper and lower bounds for $\Re\sum_ρ 1/(s-ρ)$, where the sum runs over the zeros of $ζ(s)$ on the line $1/2+it$. We also check the positivity of $\Re ξ'/ξ(s)$ for $1/2<σ<1$ assuming that there occur a non-trivial zeros of $ζ(s)$ off the critical line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_08599 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On a positivity property of the real part of logarithmic derivative of the Riemann $ξ$-function Goldštein, Edvinas Grigutis, Andrius Number Theory In this paper we investigate the positivity property of the real part of logarithmic derivative of the Riemann $ξ$-function for $1/2<σ<1$ and sufficiently large $t$. We give an explicit upper and lower bounds for $\Re\sum_ρ 1/(s-ρ)$, where the sum runs over the zeros of $ζ(s)$ on the line $1/2+it$. We also check the positivity of $\Re ξ'/ξ(s)$ for $1/2<σ<1$ assuming that there occur a non-trivial zeros of $ζ(s)$ off the critical line. |
| title | On a positivity property of the real part of logarithmic derivative of the Riemann $ξ$-function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2201.08599 |