A Non-Trivial Minoration for the Set of Salem Numbers

The set of Salem numbers is proved to be bounded from below by ${\theta }_{31}^{-1}=1.08544\dots$ where ${\theta }_n$, $n\ge 2$, is the unique root in $(0,1)$ of the trinomial $-1+x+x^n$. Lehmer’s number $1.176280\dots$ belongs to the interval $({\theta }_{12}^{-1},{\theta }_{11}^{-1})$. We conjecture that there is no Salem number in $({\theta }_{31}^{-1},{\theta }_{12}^{-1})=(1.08544\dots ,1.17295\dots )$. 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.