A proof of the Conjecture of Lehmer

The Conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions $f_{\text{house}\alpha }(z)$ associated with the dynamical zeta functions ${\zeta }_{\text{house}\alpha }(z)$ of the R'enyi–Parry arithmetical dynamical systems ($\beta$-shift), for $\alpha$ a reciprocal algebraic integer of house $\text{house}\alpha$ greater than 1, (ii) the discovery of lenticuli of poles of ${\zeta }_{\text{house}\alpha }(z)$ which uniformly equidistribute at the limit on a limit “lenticular” arc of the unit circle, when $\text{house}\alpha$ tends to $1^{+}$, giving rise to a continuous lenticular minorant ${\mathrm{M}}_r(\text{house}\alpha )$ of the Mahler measure $\mathrm{M}(\alpha )$, (iii) the Poincar'e asymptotic expansions of these poles and of this minorant ${\mathrm{M}}_r(\text{house}\alpha )$ 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 $\eta \ge 259$, 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 ${\theta }_{31}^{-1}=1.08545\dots$, 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 $\mathrm{\Theta }=1.3247\dots$, whose houses have a dynamical degree tending to infinity, the Galois orbit measures of conjugates are proved to converge towards the Haar measure on $\mathrm{\mid }z\mathrm{\mid }=1$ (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.