A proof of the Conjecture of Lehmer

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 ($\beta$-shift), for $\alpha$ a reciprocal algebraic integer 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. The Conjecture of Schinzel-Zassenhaus is proved to be true. A Dobrowolski type minoration of the Mahler measure M$(\alpha)$ is obtained. The universal minorant of M$(\alpha)$ obtained is $\theta_{\eta}^{-1} > 1$, for some integer $\eta \geq 259$, where $\theta_{\eta}$ is the positive real root of $-1+x+x^{\eta}$. 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. For sequences of algebraic integers of Mahler measure smaller than the smallest Pisot number $\Theta = 1.3247\ldots$, 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).The dynamical zeta function is used to investigate the domain of very small Mahler measures of algebraic integers in the range (1, 1.176280 . . .], if any.