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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| Accesso online: | https://arxiv.org/abs/2310.10119 |
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| _version_ | 1866911890628673536 |
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| author | Çiçek, Fatma Gonek, Steven M. |
| author_facet | Çiçek, Fatma Gonek, Steven M. |
| contents | On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros $\frac12+iγ$ of the Riemann zeta function, we show that the sequence \[ Γ_{[a, b]} =\Bigg\{ γ: γ>0 \quad \mbox{and} \quad \frac{ \log\big(| ζ^{(m_γ)} (\frac12+ iγ) | / (\logγ)^{m_γ}\big)}{\sqrt{\frac12\log\logγ}} \in [a, b] \Bigg\}, \] where the $γ$ are arranged in increasing order, is uniformly distributed modulo one. Here $a$ and $b$ are real numbers with $a<b$, and $m_γ$ denotes the multiplicity of the zero $\frac12+iγ$. The same result holds when the $γ$'s are restricted to be the ordinates of simple zeros. With an extra hypothesis, we are also able to show an equidistribution result for the scaled numbers $γ(\log T)/2π$ with $γ\in Γ_{[a, b]}$ and $0<γ\leq T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_10119 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Uniform Distribution Modulo One of Certain Subsequences of Ordinates of Zeros of the Zeta Function Çiçek, Fatma Gonek, Steven M. Number Theory On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros $\frac12+iγ$ of the Riemann zeta function, we show that the sequence \[ Γ_{[a, b]} =\Bigg\{ γ: γ>0 \quad \mbox{and} \quad \frac{ \log\big(| ζ^{(m_γ)} (\frac12+ iγ) | / (\logγ)^{m_γ}\big)}{\sqrt{\frac12\log\logγ}} \in [a, b] \Bigg\}, \] where the $γ$ are arranged in increasing order, is uniformly distributed modulo one. Here $a$ and $b$ are real numbers with $a<b$, and $m_γ$ denotes the multiplicity of the zero $\frac12+iγ$. The same result holds when the $γ$'s are restricted to be the ordinates of simple zeros. With an extra hypothesis, we are also able to show an equidistribution result for the scaled numbers $γ(\log T)/2π$ with $γ\in Γ_{[a, b]}$ and $0<γ\leq T$. |
| title | The Uniform Distribution Modulo One of Certain Subsequences of Ordinates of Zeros of the Zeta Function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2310.10119 |