A Proof of the Conjecture of Lehmer and of the Conjecture of Schinzel-Zassenhaus

Abstract

The conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions $f_{house(\alpha)}(z)$ associated with the dynamical zeta functions $\zeta_{house(\alpha)}(z)$ of the R\'enyi--Parry arithmetical dynamical systems, for $\alpha$ an algebraic integer $\alpha$ of house "$house(\alpha)$" greater than 1, (ii) the discovery of lenticuli of poles of $\zeta_{house(\alpha)}(z)$ which uniformly equidistribute at the limit on a limit "lenticular" arc of the unit circle, when $house(\alpha)$ tends to $1^+$, giving rise to a continuous lenticular minorant ${\rm M}_{r}(house(\alpha))$ of the Mahler measure ${\rm M}(\alpha)$, (iii) the Poincar\'e asymptotic expansions of these poles and of this minorant ${\rm M}_{r}(house(\alpha))$ as a function of the dynamical degree. With the same arguments the conjecture of Schinzel-Zassenhaus is proved to be true. An inequality improving those of Dobrowolski and Voutier ones is obtained. The set of Salem numbers is shown to be bounded from below by the Perron number $\theta_{31}^{-1} = 1.08545\ldots$, dominant root of the trinomial $-1 - z^{30} + z^{31}$. Whether Lehmer's number is the smallest Salem number remains open. A lower bound for the Weil height of nonzero totally real algebraic numbers, $\neq \pm 1$, is obtained (Bogomolov property). For sequences of algebraic integers of Mahler measure smaller than the smallest Pisot number, whose houses have a dynamical degree tending to infinity, the Galois orbit measures of conjugates are proved to converge towards the Haar measure on $|z|=1$ (limit equidistribution).