A Non-Trivial Minoration for the Set of Salem Numbers
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911755292114944 |
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| author | Verger-Gaugry, Jean-Louis |
| author_facet | Verger-Gaugry, Jean-Louis |
| contents | The set of Salem numbers is proved to be bounded from below by $θ_{31}^{-1}= 1.08544\ldots$ where $θ_{n}$, $ n \geq 2$, is the unique root in $(0,1)$ of the trinomial $-1+x+x^n$. Lehmer's number $1.176280\ldots$ belongs to the interval $(θ_{12}^{-1}, θ_{11}^{-1})$. We conjecture that there is no Salem number in $(θ_{31}^{-1}, θ_{12}^{-1}) = (1.08544\ldots, 1.17295\ldots)$. For proving the Main Theorem, the algebraic and analytic properties of the dynamical zeta function of the Rényi-Parry numeration system are used, with real bases running over the set of real reciprocal algebraic integers, and variable tending to 1. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_05843 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Non-Trivial Minoration for the Set of Salem Numbers Verger-Gaugry, Jean-Louis Number Theory 11K16, 11M41, 11R06, 11R09, 30B10, 30B40, 37C30, 37N99, 03D45 The set of Salem numbers is proved to be bounded from below by $θ_{31}^{-1}= 1.08544\ldots$ where $θ_{n}$, $ n \geq 2$, is the unique root in $(0,1)$ of the trinomial $-1+x+x^n$. Lehmer's number $1.176280\ldots$ belongs to the interval $(θ_{12}^{-1}, θ_{11}^{-1})$. We conjecture that there is no Salem number in $(θ_{31}^{-1}, θ_{12}^{-1}) = (1.08544\ldots, 1.17295\ldots)$. For proving the Main Theorem, the algebraic and analytic properties of the dynamical zeta function of the Rényi-Parry numeration system are used, with real bases running over the set of real reciprocal algebraic integers, and variable tending to 1. |
| title | A Non-Trivial Minoration for the Set of Salem Numbers |
| topic | Number Theory 11K16, 11M41, 11R06, 11R09, 30B10, 30B40, 37C30, 37N99, 03D45 |
| url | https://arxiv.org/abs/2401.05843 |