Zero-free regions of the Riemann zeta function and approximation in weighted Dirichlet spaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866914824942780416 |
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| author | Gallardo-Gutiérrez, Eva Seco, Daniel |
| author_facet | Gallardo-Gutiérrez, Eva Seco, Daniel |
| contents | We study zero-free regions of the Riemann zeta function $ζ$ related to an approximation problem in the weighted Dirichlet space $D_{-2}$ which is known to be equivalent to the Riemann Hypothesis since the work of Báez-Duarte. We prove, indeed, that analogous approximation problems for the standard weighted Dirichlet spaces $D_α$ when $α\in (-3,-2)$ give conditions so that the half-plane $\{s \in \mathbb{C}: \Re (s) > -\frac{α+1}{2}\}$ is also zero-free for $ζ$. Moreover, we extend such results to a large family of weighted spaces of analytic functions $\ell^p_α$. As a particular instance, in the limit case $p=1$ and $α=-2$, we provide a new equivalent formulation of the Prime Number Theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_02654 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Zero-free regions of the Riemann zeta function and approximation in weighted Dirichlet spaces Gallardo-Gutiérrez, Eva Seco, Daniel Number Theory Classical Analysis and ODEs Complex Variables Functional Analysis 11M26, 30H20 We study zero-free regions of the Riemann zeta function $ζ$ related to an approximation problem in the weighted Dirichlet space $D_{-2}$ which is known to be equivalent to the Riemann Hypothesis since the work of Báez-Duarte. We prove, indeed, that analogous approximation problems for the standard weighted Dirichlet spaces $D_α$ when $α\in (-3,-2)$ give conditions so that the half-plane $\{s \in \mathbb{C}: \Re (s) > -\frac{α+1}{2}\}$ is also zero-free for $ζ$. Moreover, we extend such results to a large family of weighted spaces of analytic functions $\ell^p_α$. As a particular instance, in the limit case $p=1$ and $α=-2$, we provide a new equivalent formulation of the Prime Number Theorem. |
| title | Zero-free regions of the Riemann zeta function and approximation in weighted Dirichlet spaces |
| topic | Number Theory Classical Analysis and ODEs Complex Variables Functional Analysis 11M26, 30H20 |
| url | https://arxiv.org/abs/2206.02654 |