A Non-Trivial Minoration for the Set of Salem Numbers

Abstract

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.

Figures (2)

• Figure 1: Curve of the Rouché condition $a \to h(1,a)$.
• Figure 2: In a) and b) the zeroes of $G_n$ and $f_{\beta}$, resp., are represented by black bullets, with lenticularity appearing (symmetrically with respect to $\mathbb{R}$) in the angular sector $|\arg z| < \pi/18$, about the unit circle in $\mathbb{C}$. In a) the first zero $z_{1,n}$ of $G_n$ (here $n=37$) is the first one in this sector having positive imaginary part. In b) the zero $\omega_{1,n}$ is obtained by a very slight deformation of $z_{1,n}$ according to Theorem \ref{['_cercleoptiSALEM']}. The other roots of $f_{\beta}$ can be found in a narrow annular neighbourhood of $|z|=1$.

Theorems & Definitions (44)

• Theorem 1.1: ex-Lehmer conjecture for Salem numbers
• Conjecture 1
• Proposition 2.1
• proof
• Definition 2.2
• Definition 2.3
• Definition 2.4: Conditions of Parry
• Definition 2.5
• Theorem 2.6
• proof