A Non-Trivial Minoration for the Set of Salem Numbers

Abstract

The set of Salem numbers is proved to be bounded from below by $\theta_{31}^{-1}= 1.08544\ldots$ where $\theta_{n}$, $ 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 $(\theta_{12}^{-1}, \theta_{11}^{-1})$. We conjecture that there is no Salem number in $(\theta_{31}^{-1}, \theta_{12}^{-1}) = (1.08544\ldots, 1.17295\ldots)$. For proving the Main Theorem, the algebraic and analytic properties of the dynamical zeta function of the R\'enyi-Parry numeration system are used, with real bases running over the set of real reciprocal algebraic integers, and variable tending to 1.