Un exemple de somme de série de vecteurs propres à valeurs propres de module un, non récurrente
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866929327603449856 |
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| author | Mazet, Pierre Saias, Eric |
| author_facet | Mazet, Pierre Saias, Eric |
| contents | Let $ζ^*(s)=\sum_{n=1}^{+\infty}(-1)^n/n^s$ and $τ$ the operator defined on the Frechet space of holomorphic functions in $\{s\in \mathbb C :1/2< Re \, s<1\}$ by $τf(s)= f(s-2iπ/\log 2)$. We show that the Riemann Hypothesis is equivalent to the strong recurrence of $ζ^*(s)$ for $τ$. It follows that a sufficient condition for $RH$ would be that every sum of a series of eigenvectors with unimodular eigenvalues for an operator $u$ is strongly recurrent for $u$. But we give a counterexample showing that it is not the case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2104_03943 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Un exemple de somme de série de vecteurs propres à valeurs propres de module un, non récurrente Mazet, Pierre Saias, Eric Complex Variables Dynamical Systems 37B20, 11M26 Let $ζ^*(s)=\sum_{n=1}^{+\infty}(-1)^n/n^s$ and $τ$ the operator defined on the Frechet space of holomorphic functions in $\{s\in \mathbb C :1/2< Re \, s<1\}$ by $τf(s)= f(s-2iπ/\log 2)$. We show that the Riemann Hypothesis is equivalent to the strong recurrence of $ζ^*(s)$ for $τ$. It follows that a sufficient condition for $RH$ would be that every sum of a series of eigenvectors with unimodular eigenvalues for an operator $u$ is strongly recurrent for $u$. But we give a counterexample showing that it is not the case. |
| title | Un exemple de somme de série de vecteurs propres à valeurs propres de module un, non récurrente |
| topic | Complex Variables Dynamical Systems 37B20, 11M26 |
| url | https://arxiv.org/abs/2104.03943 |